Appendix D Deferred proofs

This appendix proves the statements of the main text whose proofs were deferred. Each statement is restated, with its number, before its proof, together with the lemmas that only these proofs use. The subsections follow the order of the main text: optimality in its class (Appendix D.1), linear savings and product-form running times (Appendix D.2), the Steiner computations (Appendix D.3), Algorithm RS-\(\theta\) (Appendix D.4), the closure bound of Algorithm RSC (Appendix D.5), the black-box barrier (Appendix D.6), balanced set-inclusion families (Appendix D.7), the deterministic algorithms (Appendix D.8), Connected Feedback Vertex Set (Appendix D.9), the weighted finishing steps (Appendix D.10) and the further applications (Appendix D.11).

D.1 Optimality in its class

Restatement of Theorem 3.12.

Proof. (a) Let \(\mathcal G_{i-1}\) be everything the algorithm has observed before trial \(i\), including the coins it uses to choose \((t_i,\theta_i)\), and let \(i_{\max}\) be the number of trials. Every trial is charged at least \(2^0=1\), so \(i_{\max}\le \mathsf B\). The event \(\{i\le i_{\max}\}\) is decided before trial \(i\), so it is determined by \(\mathcal G_{i-1}\). Because \(\mathcal{A}\) is correct, trial \(i\) succeeds exactly when \(X_i\in\mathcal{E}_{t_i}\) and \(g(X_i)\le\theta_i\). As \(X_i\) is uniform and independent of \(\mathcal G_{i-1}\), \[\Pr\bigl[\text{trial $i$ succeeds}\bigm|\mathcal G_{i-1}\bigr] =\frac{q_{\theta_i}(t_i)}{\binom n{t_i}}\le\pi^*2^{\theta_i}.\] By the union bound and the tower property, \[\Pr[{\small\text{YES}}]\le\sum_{i\ge1}\mathbb{E}\Bigl[\mathbf 1[i\le i_{\max}]\, \Pr\bigl[\text{trial $i$ succeeds}\bigm|\mathcal G_{i-1}\bigr]\Bigr] \le\pi^*\,\mathbb{E}\Bigl[\sum_{i\le i_{\max}}2^{\theta_i}\Bigr]\le\pi^*\mathsf B .\]

(b) Let \((t,\theta)\) attain \(\pi^*\) and put \(p=q_\theta(t)/\binom nt=\pi^*2^\theta\ge 2^{\theta-\psi n}/\Gamma\). Since \(2^{\theta-\psi n}/\Gamma\le p\le1\), we have \(2^\theta\le\Gamma2^{\psi n}\), so \(\theta\le\psi n+\log\Gamma\), which is below the cap \(\Theta\ge\psi n+\log\Gamma+1\). So the algorithm runs trials of the pair \((t,\theta)\), and their number is \(m_{t,\theta}\ge M_{t,\theta}\ge\kappa\Gamma2^{\psi n-\theta}\ge\kappa/p\). Each such trial succeeds with probability \(p\), because \(\mathcal{A}\) accepts every \(X\in\mathcal{E}_t\). So all of them fail with probability at most \((1-p)^{m_{t,\theta}}\le e^{-pm_{t,\theta}}\le e^{-\kappa}\).

(c) For every \(\theta\), \[q_\theta(t)2^{-\theta}=\sum_{X\in\mathcal{E}_t,\ g(X)\le\theta}2^{-\theta} \le\sum_{X\in\mathcal{E}_t,\ g(X)\le\theta}2^{-g(X)}\le\mu_t(\mathcal{E}).\] In the integer case, with \(q_{-1}(t)=0\), exactly \(q_\theta(t)-q_{\theta-1}(t)\) sets in \(\mathcal{E}_t\) have running-time exponent \(\theta\), so \(\mu_t(\mathcal{E})=\sum_{\theta=0}^{g_{\max}}(q_\theta(t)-q_{\theta-1}(t))2^{-\theta} \le\sum_{\theta=0}^{g_{\max}}q_\theta(t)2^{-\theta}\), and the largest of these \(g_{\max}+1\) terms is at least \(\mu_t(\mathcal{E})/(g_{\max}+1)\).  ∎

D.2 Linear savings and product-form running times

This subsection proves Corollary 4.3 and the converse part of Proposition 4.5; the rest of what is known about product-form running times is in Appendix C.

D.2.1 Proof of the linear-savings corollary

We prove Corollary 4.3 here, with the notation of Section 4.2.

Restatement of Corollary 4.3.

Proof. (1) Let \(S\in\mathcal{F}_I\) with \(|S|=k\), and put \(u=2^{\rho}=1/p\). As in the proof of Theorem 4.1, Jensen’s inequality and (7) give \(\mu_t(S)\ge\binom kt2^{-(\beta n-\rho t+C)}\) for every \(t\in\{0,\dots,k\}\). Hence, by the binomial theorem, for every \(y\ge0\) \[W_S(y)\ge2^{-\beta n-C}\sum_{t=0}^{k}\binom kt(uy)^t=2^{-\beta n-C}(1+uy)^k, \qquad\text{so}\qquad \frac{(1+y)^n}{W_S(y)}\le2^{\beta n+C}\,2^{E(y)},\] where \(E(y)=n\log(1+y)-k\log(1+uy)\). The derivative \(E'(y)=\frac{1}{\ln2}\left(\frac{n}{1+y}-\frac{ku}{1+uy}\right)\) has the sign of \[n(1+uy)-ku(1+y)=n(1-\sigma u)+uy(n-k),\] which is affine in \(y\) with nonnegative slope \(u(n-k)\).

If \(\sigma\le p\), then \(\sigma u\le1\), so \(E'\ge0\) on \([0,\infty)\) and \(\min_{y\ge0}E(y)=E(0)=0\). At \(y=0\) the upper bound is therefore \(2^{\beta n+C}=2^{C+nF^*(\sigma)}\).

If \(\sigma>p\), then \(\sigma u>1\), so \(E\) decreases on \([0,y_0]\) and increases on \([y_0,\infty)\), where \(y_0=(\sigma u-1)/(u(1-\sigma))>0\) is the root of the affine function above. From \(1+y_0=(u-1)/(u(1-\sigma))\) and \(1+uy_0=\sigma(u-1)/(1-\sigma)\), \[\begin{aligned} \frac{E(y_0)}{n}&=\log(1+y_0)-\sigma\log(1+uy_0)=H(\sigma)+(1-\sigma)\log(u-1)-\log u\\ &=H(\sigma)+(1-\sigma)\log(1-p)-\sigma\rho , \end{aligned}\] where the last step uses \(\log(u-1)=\log(1-p)+\rho\) and \(\log u=\rho\). By (6) this is \(F^*(\sigma)-\beta\), so \((1+y_0)^n/W_S(y_0)\le2^{C+nF^*(\sigma)}\).

The algorithm of (1) computes a rational number \(\psi\) with \(C+nF^*(\sigma)\le\psi n\le C+nF^*(\sigma)+1\), in time \(N^{O(1)}\) from approximations of \(\beta\), \(\rho\), \(C\) and the logarithms in \(F^*(\sigma)\), and runs Algorithm MLS-\(\theta\) with rate \(\psi^+=\max(0,\psi)\), \(\Gamma=1\) and confidence \(\kappa\). In both cases above, the generating-function condition holds at \(y=0\) if \(\sigma\le p\) and at \(y=y_0\) if \(\sigma>p\), with \(\Gamma=1\) and \(2^{C+nF^*(\sigma)}\) in place of \(2^{\psi n}\), hence also with the larger value \(2^{\psi^+n}\), so (\(\star\)) holds with rate \(\psi^+\) by the first inequality of Lemma 3.9. The rate \(\psi^+\) is rational, computable and nonnegative, and \(\psi^+\le\max(0,(C+1)/n+\beta)\le N^{O(1)}\) because \(F^*(\sigma)\le\beta\) and \(n\ge1\). So Theorem 3.5 applies, and its time bound \(2^{\psi^+n}N^{O(1)}\le2^{\max(0,\,C+nF^*(\sigma)+1)}N^{O(1)}\le2\cdot2^{\max(0,\,C+nF^*(\sigma))}N^{O(1)}\) gives the claims.

(2) By the remark after (6), \(F^*(\sigma)<\beta\) holds exactly when \(\sigma>p=2^{-\rho}\), which is equivalent to \(\rho>\log(1/\sigma)\).  ∎

D.2.2 Proof of the converse for product-form running times

We prove here part (c) of Proposition 4.5 and the final inequalities stated after it; parts (a) and (b) are proved in Section 4.3. We keep the notation of that proposition.

Restatement of Proposition 4.5.

Proof of Proposition 4.5(c) and of the final inequalities. (c) Put \(u=bc/d\) and, for real \(x\in[0,n]\) and \(y\ge0\), \(E_x(y)=n\log(1+y)-x\log(1+uy)\), so that by (a) \((1+y)^n/W(y)=b^nc^k2^{E_k(y)}\) for integers \(k\). As in the proof of Corollary 4.3, the derivative \(E_x'(y)\) has the sign of \(n(1-(x/n)u)+uy(n-x)\).

The case \(c=1\). For \(k=0\) only \(X=\emptyset\) occurs: \(W(y)=b^{-n}\), so \(G_0=\inf_{y\ge0}b^n(1+y)^n=b^n\), attained at \(y=0\), and \(M_0=b^n\). Also \(\mathrm{base}(b,1,d)=b\).

The case \(c>1\) and \(d<bc\). Here \(\sigma^*>0\), and \(\sigma^*<1\) because \(cd<bc+cd-d\) is equivalent to \(d<bc\). Moreover \(\sigma^*u=bc^2/(bc+cd-d)>1\), because this is equivalent to \(bc^2-bc>cd-d\), that is \(bc(c-1)>d(c-1)\), which holds as \(c>1\) and \(d<bc\). Take \(x=\sigma^*n\). Then \(E_x'(0)<0\), and since \(x<n\) the affine function that carries the sign of \(E_x'\) has positive slope, so \(E_x\) has a unique minimizer on \([0,\infty)\), the root of \(n(1+uy)=xu(1+y)\). Since \(1+uy^*=c\) and \(\sigma^*u(1+y^*)=\frac{bc^2}{bc+cd-d}\cdot\frac{bc+cd-d}{bc}=c\), the root is \(y^*\). Therefore \[\tag{24} \inf_{y\ge0}\Bigl[n\log\bigl(b(1+y)\bigr)+x\log\frac{c}{1+uy}\Bigr] =n\log\bigl(b(1+y^*)\bigr)+x\log\frac{c}{c}=n\log\Bigl(b+d-\frac dc\Bigr),\] and if \(\sigma^*n\) is an integer \(k\), the left-hand side is \(\log G_k\), so \(G_k=(b+d-d/c)^n\).

For general \(n\ge1\), define for real \(x\in[0,n]\) \[\mathsf f(x)=\inf_{y\ge0}\Bigl[n\log\bigl(b(1+y)\bigr)+x\log\frac{c}{1+uy}\Bigr],\] so that \(\mathsf f(k)=\log G_k\) for integers \(0\le k\le n\), by (a). The function \(\mathsf f\) is finite: since \(u>1\) we have \(1+uy\le u(1+y)\), so the bracket is at least \(n\log b+(n-x)\log(1+y)+x\log(c/u)\ge n\log b-n|\log(c/u)|\). As an infimum of functions that are affine in \(x\), \(\mathsf f\) is concave on \([0,n]\). We have \(\mathsf f(0)=n\log b\) (the bracket is \(n\log(b(1+y))\), minimized at \(y=0\)), and \(\mathsf f(\sigma^*n)=n\log(b+d-d/c)\) by (24); note that \(b+d-d/c\ge b\), because \(c\ge1\). Let \(k=\lfloor\sigma^*n\rfloor\), so \(\sigma^*n-1<k\le\sigma^*n\). By concavity on \([0,\sigma^*n]\), \[\begin{aligned} \log G_k=\mathsf f(k)&\ge\Bigl(1-\frac{k}{\sigma^*n}\Bigr)\mathsf f(0)+\frac{k}{\sigma^*n}\,\mathsf f(\sigma^*n) =n\log b+\frac{k}{\sigma^*}\log\frac{b+d-d/c}{b}\\ &\ge n\log\Bigl(b+d-\frac dc\Bigr)-\frac{1}{\sigma^*}\log\frac{b+d-d/c}{b}, \end{aligned}\] which is the claimed bound.

The case \(c>1\) and \(d\ge bc\). Take \(k=n\). Now \(u\le1\), so \(1+uy\le1+y\) and \[\frac{(1+y)^n}{W(y)}=(bc)^n\Bigl(\frac{1+y}{1+uy}\Bigr)^n\ge(bc)^n,\] with equality at \(y=0\); so \(G_n=(bc)^n\). Also \(M_n=\min_{0\le t\le n}(bc)^{n-t}d^t=(bc)^n\), attained at \(t=0\), since \(d\ge bc\).

The final inequalities. The upper bound \(\max_kM_k\le \mathrm{base}(b,c,d)^n\) follows from (b). For the lower bound, Lemma 3.9 gives \(M_k\ge G_k/(n+1)\) for every \(k\), and the three cases give some \(k\) with \(G_k\ge \mathrm{base}(b,c,d)^n/\mathsf K\): in the second case \(\mathrm{base}(b,c,d)=b+d-d/c\) because \(d<bc\).  ∎

D.3 The Steiner computations

This subsection proves the statements of Sections 5.1 and 5.2, with the notation of those subsections.

Restatement of Lemma 5.3.

Proof. If \(K=\emptyset\), then \(D=\emptyset\) is feasible and the answer is \(0\); let \(K\neq\emptyset\). Let \(\varphi'\) agree with \(\varphi\) outside \(\{r\}\cup K\) and vanish on \(\{r\}\cup K\). Give every arc \(uv\) the length \(\mathrm{len}(uv)=n\,\varphi'(v)+1\), a positive integer.

Branching walks. A branching walk from \(v\) is a finite sequence \(\bigl((u_1,\mathcal W_1),\dots,(u_q,\mathcal W_q)\bigr)\), \(q\ge0\), in which \(vu_i\in A(\vec G)\) and \(\mathcal W_i\) is a branching walk from \(u_i\), for every \(i\); this is an inductive definition, and the empty sequence is the branching walk with \(q=0\). Its image is \(\{v\}\) together with the images of the \(\mathcal W_i\), its arcs are the arcs \(vu_i\) together with the arcs of the \(\mathcal W_i\) (counted with multiplicity), and its length is the sum of the lengths of its arcs. Equivalently, a branching walk is a rooted ordered tree mapped into \(\vec G\) so that the root goes to \(v\) and every tree edge, directed away from the root, goes to an arc of \(\vec G\). By induction, every vertex of the image other than \(v\) is the head of one of its arcs, and every vertex of the image is reachable from \(v\) in the subdigraph induced by the image.

For \(Y\subseteq V(\vec G)\), \(v\in V(\vec G)\) and an integer \(\mathsf L\ge0\) let \(b_Y(v,\mathsf L)\) be the number of branching walks from \(v\) of length \(\mathsf L\) whose image is disjoint from \(Y\). Then \(b_Y(v,\mathsf L)=0\) if \(v\in Y\), and for \(v\notin Y\) \[b_Y(v,0)=1,\qquad b_Y(v,\mathsf L)=\sum_{\substack{vu\in A(\vec G)\\u\notin Y}}\ \sum_{\mathsf L_1=0}^{\mathsf L-\mathrm{len}(vu)}b_Y(u,\mathsf L_1)\,b_Y\bigl(v,\mathsf L-\mathrm{len}(vu)-\mathsf L_1\bigr) \quad(\mathsf L\ge1).\] Indeed, the only branching walk of length \(0\) is the empty one, since every arc has positive length. A nonempty branching walk from \(v\) splits uniquely into its first pair \((u_1,\mathcal W_1)\) and the branching walk formed by the remaining pairs, which again starts at \(v\); the length is \(\mathrm{len}(vu_1)\) plus the two lengths, and the image avoids \(Y\) exactly when \(u_1\notin Y\) and both pieces avoid \(Y\) (the image of the remainder contains \(v\)). The right-hand side refers only to lengths below \(\mathsf L\), so the table can be filled in order of increasing \(\mathsf L\).

Inclusion–exclusion. For \(\mathsf L\ge0\) put \(b(\mathsf L)=\sum_{Y\subseteq K}(-1)^{|Y|}b_Y(r,\mathsf L)\). A branching walk from \(r\) with image \(\mathcal I\) is counted in \(b_Y(r,\mathsf L)\) exactly for the sets \(Y\subseteq K\setminus \mathcal I\), and \(\sum_{Y\subseteq K\setminus \mathcal I}(-1)^{|Y|}\) is \(1\) if \(K\subseteq \mathcal I\) and \(0\) otherwise. So \(b(\mathsf L)\) is the number of branching walks from \(r\) of length \(\mathsf L\) whose image contains \(K\).

Least length. We show that \(b(\mathsf L)>0\) for some \(\mathsf L\) if and only if \(\operatorname{OPT}=\operatorname{OPT}_\varphi(\vec G,r,K)<\infty\), and that the least such \(\mathsf L\), say \(\mathsf L^*\), satisfies \(\lfloor \mathsf L^*/n\rfloor=\operatorname{OPT}\). First let a branching walk from \(r\) have image \(\mathcal I\supseteq K\) and length \(\mathsf L\). Every vertex of \(\mathcal I\setminus\{r\}\) is the head of one of its arcs, and the arcs have lengths at least \(n\,\varphi'\) of their heads, so, as \(\varphi'\ge0\) and \(\varphi'(r)=0\), \[\mathsf L\ \ge\ n\sum_{v\in \mathcal I}\varphi'(v)\ =\ n\,\varphi(D_{\mathcal I}),\qquad D_{\mathcal I}=\mathcal I\setminus(\{r\}\cup K).\] Every terminal is reachable from \(r\) in \(\vec G[\mathcal I]=\vec G[\{r\}\cup K\cup D_{\mathcal I}]\), so \(D_{\mathcal I}\) is feasible and \(\lfloor \mathsf L/n\rfloor\ge \varphi(D_{\mathcal I})\ge\operatorname{OPT}\). Conversely, let \(D\) be feasible with \(\varphi(D)=\operatorname{OPT}\), and let \(T\) be an out-tree of \(\vec G[\{r\}\cup K\cup D]\) rooted at \(r\) that spans the vertices reachable from \(r\) there, for instance a breadth-first search tree. Ordering the children of every vertex arbitrarily turns \(T\) into a branching walk from \(r\) whose image contains \(K\). It has \(|A(T)|\le n-1\) arcs, each non-root vertex of \(T\) is the head of exactly one of them, and the non-root vertices of \(T\) outside \(K\) lie in \(D\); so its length is at most \(n\,\varphi(D)+|A(T)|\le n\,\operatorname{OPT}+n-1\). Hence \(\mathsf L^*\) exists and \(\lfloor \mathsf L^*/n\rfloor\le\operatorname{OPT}\), and with the first part \(\lfloor \mathsf L^*/n\rfloor=\operatorname{OPT}\). As \(\operatorname{OPT}\le \varphi(V(\vec G))\le n\,\varphi_{\max}\), it suffices to compute \(b(\mathsf L)\) for \(0\le \mathsf L\le \mathsf L_{\max}:=n^2\varphi_{\max}+n-1\); if all these values are \(0\), then \(\operatorname{OPT}=\infty\).

Running time and space. A branching walk of length at most \(\mathsf L\) has at most \(\mathsf L\) arcs; it is determined by a rooted ordered tree with \(e\le \mathsf L\) edges, of which there are at most \(4^e\), and by the images of the \(e\) non-root vertices. So \(b_Y(v,\mathsf L)\le(\mathsf L+1)(4n)^{\mathsf L}\), and all numbers involved, including the partial sums of \(b(\mathsf L)\), have \(O(\mathsf L_{\max}\log n+n)\) bits, which is polynomial in \(n+\varphi_{\max}\). For one set \(Y\) the table \(\bigl(b_Y(v,\mathsf L)\bigr)_{v,\,\mathsf L\le \mathsf L_{\max}}\) has \(n(\mathsf L_{\max}+1)\) entries, each computed with \(O(n\mathsf L_{\max})\) arithmetic operations. The algorithm runs through the \(2^{|K|}\) sets \(Y\subseteq K\) one at a time, keeps one table and the \(\mathsf L_{\max}+1\) partial sums of \(b(\mathsf L)\), and finally returns \(\lfloor \mathsf L^*/n\rfloor\) or \(\infty\). This takes time \(2^{|K|}(n+\varphi_{\max})^{O(1)}\) and space \((n+\varphi_{\max})^{O(1)}\).  ∎

Restatement of Lemma 5.6.

Proof. Let \(C_1\in\mathcal{D}\) and put \(\rho=\hat C_1\) and \(K=K_1\). The non-root, non-terminal vertices of \(\vec G_J\) are the vertices \(\hat b\) with \(b\in B\), so a feasible set for \((\vec G_J,\rho,K)\) in the sense of Section 5.1 has the form \(\hat Z=\{\hat b:b\in Z\}\) with \(Z\subseteq B\), and its weight is \(\varphi(Z)\). We show that \[\tag{25} \operatorname{OPT}_\varphi(\vec G_J,\rho,K)=\min\bigl\{\varphi(Y):Y\text{ a solution of }J\bigr\},\] which is the claim.

From solutions to feasible sets. Let \(Y\) be a solution of \(J\). Let \(H\) be the graph obtained from the connected graph \(G[V(\mathcal{D})\cup Y]\) by contracting each member of \(\mathcal{D}\) into one vertex; it is connected, and its vertices correspond to the vertices \(\hat C\) (\(C\in\mathcal{D}\)) and \(\hat y\) (\(y\in Y\)) of \(\vec G_J\), among them \(\rho\). Every edge of \(H\) joins two vertices \(y,y'\in Y\subseteq B\) or a vertex \(C\in\mathcal{D}\) to a vertex \(y\in N(C)\cap Y\), since distinct members of \(\mathcal{D}\) are non-adjacent; in both cases \(\vec G_J\) has the two opposite arcs between the corresponding vertices. So every \(\hat C\) (\(C\in\mathcal{D}\)) and every \(\hat y\) (\(y\in Y\)) is reachable from \(\rho\) in \(\vec G_J[\{\rho\}\cup K\cup\hat Y]\). Every \(r\in R\) lies in \(N[Y]\), so some \(y\in Y\cap N[r]\subseteq N[r]\cap B\) exists, \(\hat y\) is reachable from \(\rho\), and \(\check r\) is reached through the arc \(\hat y\check r\). Hence \(\hat Y\) is feasible, of weight \(\varphi(Y)\), and the left-hand side of (25) is at most the right-hand side.

From feasible sets to solutions. Let \(\hat Z\) be feasible, let \(Q\) be the set of vertices reachable from \(\rho\) in \(\vec G_J[\{\rho\}\cup K\cup\hat Z]\), which contains \(\rho\) and \(K\), and let \(Y'=\{b\in Z:\hat b\in Q\}\). Then \(Y'\subseteq B\) and \(\varphi(Y')\le\varphi(Z)\). The vertices of \(\check R\) have no outgoing arcs, so every vertex of \(Q\setminus\check R\) is reached from \(\rho\) by a path all of whose vertices lie in \(Q\setminus\check R\), that is, are of the form \(\hat C\) (\(C\in\mathcal{D}\)) or \(\hat b\) (\(b\in Y'\)). Each arc of such a path joins two vertices whose underlying sets (\(C\) or \(\{b\}\)) are joined by an edge of \(G\), and every \(C\in\mathcal{D}\) induces a connected subgraph of \(G\); as \(Q\setminus\check R\) consists of all \(\hat C\) (\(C\in\mathcal{D}\)) and all \(\hat b\) (\(b\in Y'\)), the graph \(G[V(\mathcal{D})\cup Y']\) is connected. It is nonempty, because it contains \(C_1\). Every \(\check r\) with \(r\in R\) lies in \(Q\), and its predecessor on a path from \(\rho\) is some \(\hat b\in Q\) with \(b\in N[r]\cap B\), so \(b\in Y'\); hence \(R\subseteq N[Y']\). So \(Y'\) is a solution of \(J\), and the right-hand side of (25) is at most \(\varphi(Y')\le\varphi(Z)\). This proves (25).

For the running time, let first \(\mathcal{D}\neq\emptyset\). The digraph \(\vec G_J\) is built in polynomial time, has at most \(2n\) vertices and weights at most \(\varphi_{\max}\), and Lemma 5.3 is applied once, with \(|K_1|=|\mathcal{D}|-1+|R|=|\mathrm{T}(J)|-1\) terminals; this takes time \(2^{|\mathrm{T}(J)|}(n+\varphi_{\max})^{O(1)}\) and space \((n+\varphi_{\max})^{O(1)}\), deterministically. If \(\mathcal{D}=\emptyset\), every solution \(Y\) of \(J\) is nonempty, because \(V(\mathcal{D})\cup Y\neq\emptyset\). For \(b\in B\) put \(J^{(b)}=(\{\{b\}\},B\setminus\{b\},R\setminus N[b])\). This is a state (\(\{b\}\) is connected, \(B\setminus\{b\}\) avoids it and \(R\setminus N[b]\) avoids \(N[b]\)), and a set \(Y\subseteq B\) with \(b\in Y\) is a solution of \(J\) if and only if \(Y\setminus\{b\}\) is a solution of \(J^{(b)}\): \(G[Y]=G[\{b\}\cup(Y\setminus\{b\})]\), and \(R\subseteq N[Y]=N[b]\cup N[Y\setminus\{b\}]\) holds if and only if \(R\setminus N[b]\subseteq N[Y\setminus\{b\}]\). Hence \(\operatorname{OPT}_\varphi(J)=\min_{b\in B}\bigl(\varphi(b)+\operatorname{OPT}_\varphi(J^{(b)})\bigr)\), with \(\min\emptyset=\infty\), and \(|\mathrm{T}(J^{(b)})|=|R\setminus N[b]|+1\le|\mathrm{T}(J)|+1\). Applying the first part to the at most \(n\) states \(J^{(b)}\) gives the same bounds.  ∎

Restatement of Corollary 5.8.

Proof. The first statement is Lemma 5.6 with \(\varphi\equiv1\), so that \(\varphi_{\max}=1\) and the bounds become \(2^{|\mathrm{T}(J)|}n^{O(1)}\) and \(n^{O(1)}\).

For the second statement, \(\Phi_{\mathrm{CDS}}\) is an implicit set system (Definition 3.1): \(V(G)\) is listed and connected domination is tested in polynomial time, and strings that do not encode a graph get an empty universe. By the description of the solutions of \(J_X\) after (12), \(\mathcal{A}_{\mathrm{St}}\) accepts \((G,X,k)\) exactly when some \(Y\subseteq V\setminus X\) with \(|Y|\le k-|X|\) makes \(X\cup Y\) a connected dominating set. Its running time on every input, whether or not \(X\) lies in a solution, is \(2^{|\mathrm{T}(J_X)|}n^{O(1)}\) by the first statement, and \(|\mathrm{T}(J_X)|\) is computed in polynomial time by (13), which also gives the range.  ∎

D.4 Algorithm RS-\(\theta\)

This subsection proves the statements of Sections 7.2, 7.3 and 7.4, with the notation of Section 7, and the lower bound of Section 7.6 on uniform lobsters (Appendix D.4.3).

D.4.1 The tree reduction and the three-state recursion

Restatement of Lemma 7.3.

Proof. By (13), \(|\mathrm{T}(J_X)|=|(V\setminus S)\setminus N[X]|+|S\setminus N[X]|+\operatorname{comp}(G[X])\), and we bound its three parts by the three terms of \(\Pi_{F,\nu}(X)\) in turn. (1) If \(u\in V\setminus S\) is not in \(N[X]\), then its neighbor \(o(u)\) is not in \(X\); so \(|(V\setminus S)\setminus N[X]|\le\sum_{v\in S\setminus X}|o^{-1}(v)|=\sum_{v\in S\setminus X}\nu_v\). (2) If \(v\in S\) is not in \(N[X]\), then \(N_G[v]\cap X=\emptyset\), so \(N_F[v]\cap X=\emptyset\) because \(N_F[v]\subseteq N_G[v]\). (3) \(F[X]\) is a spanning subgraph of \(G[X]\), as \(X\subseteq S=V(F)\); so \(\operatorname{comp}(G[X])\le\operatorname{comp}(F[X])\).

For the equality case, suppose that \(o(u)\) is the only neighbor of \(u\) in \(S\) for every \(u\notin S\). Then, as \(X\subseteq S\), \(u\notin N[X]\) if and only if \(o(u)\notin X\), so (1) is an equality. If moreover \(G[S]=F\), then \(N_G[v]\cap S=N_F[v]\) for \(v\in S\) and \(G[X]=F[X]\), so (2) and (3) are equalities. The last claim holds termwise, as \(2^{-|\mathrm{T}(J_X)|}\ge2^{-\Pi_{F,\nu}(X)}\).  ∎

Restatement of Lemma 7.4.

Proof. The proof is a double induction whose core is a decomposition of the potential of a partial subtree into the potentials of its two pieces, with one correction for the edge joining \(v\) to the new child and one for the domination of that child; the three states carry the information that these corrections need.

Let \(F'\) be the subtree consisting of a vertex \(v\) and the subtrees \(F_{w_1},\dots,F_{w_i}\) of its first \(i\) children (\(0\le i\le d\)). For \(X\subseteq V(F')\) let \[\Pi^{\mathrm{in}}_{F'}(X)=\sum_{u\in V(F')\setminus X}\nu_u+\bigl|\{u\in V(F')\setminus\{v\}:N_F[u]\cap X=\emptyset\}\bigr| +\operatorname{comp}(F'[X]);\] for \(u\neq v\) in \(F'\) the set \(N_F[u]\) lies inside \(V(F')\), since the parent of \(u\) is \(v\) or lies in the same subtree \(F_{w_k}\) for some \(k\le i\). Let \(P'=(P'_1,P'_2,P'_3)\) be the sums of \(y^{|X|}2^{-\Pi^{\mathrm{in}}_{F'}(X)}\) over the sets \(X\subseteq V(F')\) in states \(1\), \(2\) and \(3\) of \(v\), where “child” means one of \(w_1,\dots,w_i\). We show, by induction on the height of \(v\) and then on \(i\), that \(P'=\bigl(\cdots(\mathrm{Init}_y(\nu_v)\circ\mathrm{msg}_{w_1})\circ\cdots\bigr)\circ\mathrm{msg}_{w_i}\); for \(i=d\) this identifies \(\mathrm{msg}_v\) as the triple of \(F_v\), whose entries are nonnegative. Write \(\Pi^{\mathrm{in}}_{F_v}\) for \(\Pi^{\mathrm{in}}_{F'}\) when \(F'=F_v\).

For \(i=0\), \(F'\) consists of \(v\) alone. The set \(X=\{v\}\) has \(\Pi^{\mathrm{in}}_{F'}(X)=1\) (one component, no owner term) and \(X=\emptyset\) has \(\Pi^{\mathrm{in}}_{F'}(X)=\nu_v\), and state \(2\) is empty because \(v\) has no children yet; so the triple is \((y/2,0,2^{-\nu_v})=\mathrm{Init}_y(\nu_v)\). For the step from \(i-1\) to \(i\), let \(F''\) be the subtree for \(i-1\), with triple \(P''\) and potential \(\Pi^{\mathrm{in}}_{F''}\), let \(w=w_i\), and let \(Q=\mathrm{msg}_w\) be the triple of \(F_w\), by the induction hypothesis. Every \(X\subseteq V(F')\) splits uniquely as \(X=X''\cup X_w\) with \(X''\subseteq V(F'')\) and \(X_w\subseteq V(F_w)\), and \[\begin{split} \Pi^{\mathrm{in}}_{F'}(X)&=\Pi^{\mathrm{in}}_{F''}(X'')+\Pi^{\mathrm{in}}_{F_w}(X_w)-[v\in X\text{ and }w\in X]\\ &\quad+[v\notin X,\ w\notin X\text{ and no child of $w$ is in }X]. \end{split}\] Indeed, the owner terms are additive over the split. A vertex \(u\notin\{v,w\}\) has its closed neighborhood inside \(F''\) or inside \(F_w\), so it is counted in \(\Pi^{\mathrm{in}}_{F'}\) exactly when it is counted in \(\Pi^{\mathrm{in}}_{F''}\) or \(\Pi^{\mathrm{in}}_{F_w}\). The vertex \(w\) is not counted in \(\Pi^{\mathrm{in}}_{F_w}\), and it is counted in \(\Pi^{\mathrm{in}}_{F'}\) exactly when \(N_F[w]\cap X=\emptyset\), that is, when \(v\), \(w\) and the children of \(w\) are not in \(X\); the vertex \(v\) is counted in neither. Finally, \(F'[X]\) is the disjoint union of \(F''[X'']\) and \(F_w[X_w]\), together with the edge \(vw\) if \(v,w\in X\), which joins two components into one.

Now we sum the weights over the states of \(v\) and \(w\). If \(v\in X\), then \(w\) contributes the factor \(2Q_1\) if \(w\in X\) (one component fewer) and \(Q_2+Q_3\) otherwise (\(w\) is dominated by \(v\)); so \(P'_1=P''_1(2Q_1+Q_2+Q_3)\). If \(v\) is in state \(2\) for \(F''\), then \(v\) stays in state \(2\), and \(w\) contributes \(Q_1\), \(Q_2\) or \(Q_3/2\) according to its state (in state \(3\), \(w\) is not dominated, which adds \(1\) to the potential and so multiplies the weight by \(\frac12\)). If \(v\) is in state \(3\) for \(F''\), then \(w\in X\) moves \(v\) to state \(2\) (term \(P''_3Q_1\)), and otherwise \(v\) stays in state \(3\), with the factor \(Q_2+Q_3/2\). So \(P'=P''\circ Q\).

At the root, \(N_F[r]\) consists of \(r\) and its children, so \(\Pi_{F,\nu}(X)=\Pi^{\mathrm{in}}_{F_r}(X)+[r\text{ is in state }3]\), and \(W_{F,\nu}(y)=\mathrm{msg}_{r,1}+\mathrm{msg}_{r,2}+\mathrm{msg}_{r,3}/2=\mathrm{fin}(\mathrm{msg}_r)\).  ∎

D.4.2 Antinorm certificates in the cumulative cone

We collect the facts about \(\mathcal K_{\mathrm{cum}}\) that the certificate argument uses, and then the abstract mechanism, which Section 8 reuses with another cone.

Lemma D.4.1 (Closure facts). 

(a)  If \(Q\in\mathbb{R}^3_{\ge0}\) and \(\Delta\in\mathcal K_{\mathrm{cum}}\), then \(\Delta\circ Q\in\mathcal K_{\mathrm{cum}}\) and \(Q\circ \Delta\in\mathcal K_{\mathrm{cum}}\).

(b)  \(\mathrm{fin}(\Delta)\ge0\) for every \(\Delta\in\mathcal K_{\mathrm{cum}}\).

Lean proof: merge_Kcum_left, merge_Kcum_right, fin_nonneg_of_Kcum, PZ_add_Kcum, smul_PZ_add_Kcum, smul_PZ_mono (formalisation)

Proof. (a) The operation \(\circ\) is bilinear and \(\mathcal K_{\mathrm{cum}}\) is a convex cone, so it suffices to check the three generators of \(\mathcal K_{\mathrm{cum}}\). For \(Q\ge0\) we list each product and its cumulative coordinates: \((1,-1,0)\circ Q=\bigl(2Q_1+Q_2+Q_3,\,-Q_1-Q_2-\tfrac{Q_3}2,\,0\bigr)\), with cumulative coordinates \(\bigl(2Q_1+Q_2+Q_3,\,Q_1+\tfrac{Q_3}2,\,Q_1+\tfrac{Q_3}2\bigr)\); \((0,1,-1)\circ Q=\bigl(0,\,Q_2+\tfrac{Q_3}2,\,-Q_2-\tfrac{Q_3}2\bigr)\), with \(\bigl(0,\,Q_2+\tfrac{Q_3}2,\,0\bigr)\); and \((0,0,1)\circ Q=\bigl(0,\,Q_1,\,Q_2+\tfrac{Q_3}2\bigr)\ge0\). In the other order, \(Q\circ(1,-1,0)=(Q_1,\,Q_3,\,-Q_3)\), with cumulative coordinates \((Q_1,\,Q_1+Q_3,\,Q_1)\); \(Q\circ(0,1,-1)=\bigl(0,\,\tfrac{Q_2}2,\,\tfrac{Q_3}2\bigr)\ge0\); and \(Q\circ(0,0,1)=\bigl(Q_1,\,\tfrac{Q_2}2,\,\tfrac{Q_3}2\bigr)\ge0\). All cumulative coordinates are nonnegative. (b) \(\mathrm{fin}(\Delta)=\tfrac12(\Delta_1+\Delta_2)+\tfrac12(\Delta_1+\Delta_2+\Delta_3)\), and both brackets are nonnegative for \(\Delta\in\mathcal K_{\mathrm{cum}}\).  ∎

The nonnegativity of the other factor in (a) is needed: \((1,-1,0)\circ(0,1,-1)=(0,-\tfrac12,0)\notin\mathcal K_{\mathrm{cum}}\).

The next lemma isolates the mechanism: a convex region that the merge operation maps into itself up to a known scale, for any bilinear operation and any cone with the closure property of Lemma D.4.1(a).

Lemma D.4.2 (Antinorm certificates for a bilinear merge).  Let \(\circ\) be a bilinear operation on \(\mathbb{R}^m\), let \(\mathcal K\subseteq\mathbb{R}^m\) be a convex cone containing \(\mathbb{R}^m_{\ge0}\) such that \(\Delta\circ Q\in\mathcal K\) and \(Q\circ\Delta\in\mathcal K\) for every \(\Delta\in\mathcal K\) and every \(Q\in\mathbb{R}^m_{\ge0}\), and for a finite nonempty set \(Z\subseteq\mathbb{R}^m_{\ge0}\) put \(\mathcal P_Z=\mathrm{conv}(Z)+\mathcal K\). Then:

(a)  \(\mathcal P_Z+\mathcal K=\mathcal P_Z\), and \(\eta\mathcal P_Z\subseteq\eta'\mathcal P_Z\) whenever \(0<\eta'\le\eta\);

(b)  let \(Y,Y',Z\subseteq\mathbb{R}^m_{\ge0}\) be finite and nonempty with \(y\circ z\in\mathcal P_{Y'}\) for all \(y\in Y\) and \(z\in Z\); if \(P\in\eta\mathcal P_Y\) and \(Q\in\eta'\mathcal P_Z\) with \(\eta,\eta'>0\) and \(Q\ge0\), then \(P\circ Q\in\eta\eta'\mathcal P_{Y'}\).

Proof. (a) The first claim holds because \(\mathcal K+\mathcal K=\mathcal K\). For the second, let \(P=\eta(z+\Delta)\) with \(z\in\mathrm{conv}(Z)\) and \(\Delta\in\mathcal K\). Then \(P=\eta'(z+\Delta'')\) with \(\Delta''=\frac{\eta-\eta'}{\eta'}z+\frac\eta{\eta'}\Delta\), and \(\Delta''\in\mathcal K\) because \(z\ge0\) and \(\mathcal K\supseteq\mathbb{R}^m_{\ge0}\) is a convex cone.

(b) Write \(P=\eta(y+\Delta)\) and \(Q=\eta'\hat Q\) with \(\hat Q=z+\Delta'\ge0\), where \(y\in\mathrm{conv}(Y)\), \(z\in\mathrm{conv}(Z)\) and \(\Delta,\Delta'\in\mathcal K\). By bilinearity, \[\frac{P\circ Q}{\eta\eta'}=y\circ z+y\circ \Delta'+\Delta\circ\hat Q .\] With convex combinations \(y=\sum_i\alpha_iy_i\) and \(z=\sum_j\beta_jz_j\) of points of \(Y\) and of \(Z\) we get \(y\circ z=\sum_{i,j}\alpha_i\beta_j\,y_i\circ z_j\in\mathcal P_{Y'}\), as \(\mathcal P_{Y'}\) is convex. Since \(y\ge0\) and \(\hat Q\ge0\), the closure property gives \(y\circ \Delta'\in\mathcal K\) and \(\Delta\circ\hat Q\in\mathcal K\), and (a) gives \(P\circ Q\in\eta\eta'\mathcal P_{Y'}\).  ∎

Restatement of Lemma 7.5.

Proof. Root \(F\) anywhere and let \(\mathrm{msg}_v\) be the messages of Lemma 7.4, which are nonnegative. By induction from the leaves, \(\mathrm{msg}_v\in e_1^{|V(F_v)|}e_2^{\,\nu(F_v)}\mathcal P_Z\), where \(\nu(F_v)=\sum_{u\in V(F_v)}\nu_u\): the start \(\mathrm{Init}_y(\nu_v)\) lies in \(e_1e_2^{\,\nu_v}\mathcal P_Z\) by (b), and each merge with a child \(w\) multiplies the scale by \(e_1^{|V(F_w)|}e_2^{\,\nu(F_w)}\), by Lemma D.4.2(b) with \(Y=Y'=Z\), applied with \(Q=\mathrm{msg}_w\ge0\), whose closure hypothesis is Lemma D.4.1(a) and whose hypothesis on \(Z\) is (a). At the root, \(\mathrm{msg}_r=\eta(z+\Delta)\) with \(\eta=e_1^se_2^{\nu(F)}\), \(z\in\mathrm{conv}(Z)\) and \(\Delta\in\mathcal K_{\mathrm{cum}}\). Since \(\mathrm{fin}\) is linear, Lemma 7.4 and Lemma D.4.1(b) give \[W_{F,\nu}(y)=\mathrm{fin}(\mathrm{msg}_r)=\eta\bigl(\mathrm{fin}(z)+\mathrm{fin}(\Delta)\bigr)\ge\eta\,\mathrm{fin}(z)\ge\eta\,c_Z .\] For the last claim let \(j>j_0\). Then \(\mathrm{Init}_y(j)-(y/2,0,0)=(0,0,2^{-j})\in\mathcal K_{\mathrm{cum}}\), so by Lemma D.4.2(a) \(\mathrm{Init}_y(j)\in e_1e_2^{\,j_0+1}\mathcal P_Z+\mathcal K_{\mathrm{cum}}=e_1e_2^{\,j_0+1}\mathcal P_Z\subseteq e_1e_2^{\,j}\mathcal P_Z\), because \(e_1e_2^{\,j}\le e_1e_2^{\,j_0+1}\).  ∎

Restatement of Corollary 7.6.

Proof. Since \(G[S]\) is connected, it has a spanning tree \(F\); since \(S\) is dominating, an owner map exists, and its counts satisfy \(\sum_v\nu_v=n-s\). Lemmas 7.3 and 7.5 give the first inequality. The second is the first inequality of Lemma 3.9.  ∎

Remark D.4.3 (Why the cumulative cone).  By Lemma D.4.1(a) and (b), moving weight from state \(2\) to state \(1\), or from state \(3\) to state \(2\), never decreases the final value \(\mathrm{fin}(\mathrm{msg}_r)\). The nonnegative orthant is also preserved by \(\circ\), so Lemma 7.5 holds verbatim with \(\mathbb{R}^3_{\ge0}\) in place of \(\mathcal K_{\mathrm{cum}}\). The cumulative cone is larger: in its order a deficit in state \(3\) can be compensated by weight in states \(1\) or \(2\) (an attempt with the orthant is reported in Appendix F.2).

D.4.3 A lower bound on uniform lobsters

We prove here the following proposition of Section 7.6, with the notation of that section.

Restatement of Proposition 7.9.

Proof. (a) Let \(S\) be a connected dominating set of \(\Lambda_L\), where \(L\ge1\), so that \(n\ge7\). If a leg \(u\) is not in \(S\), then its two claws, whose only neighbor is \(u\), must lie in \(S\) to be dominated; they are not adjacent and every path between them passes through \(u\), contradicting the connectivity of \(G[S]\). So all legs lie in \(S\). The neighbors of a leg of \(x_i\) are \(x_i\) and its two claws, which have no other neighbors, and \(\Lambda_L\) has at least two legs; so a path in \(G[S]\) from a leg of \(x_i\) to another leg passes through \(x_i\), and \(x_i\in S\). Hence \(S\supseteq S^*\). Conversely, \(S^*\) induces a tree (a caterpillar: \(\Lambda_L\) without its claws) and dominates every claw through its leg, so it is a connected dominating set.

(b) By (a), the only connected dominating set of size at most \(s=\gamma_{\mathrm{c}}(\Lambda_L)\) is \(S^*\). A trial accepts only if \(\mathcal{A}_{\mathrm{St}}\) is run on a sample \(X\) for which some \(Y\) with \(|Y|\le s-t\) makes \(X\cup Y\) a connected dominating set, so only if \(X\cup Y=S^*\) and in particular \(X\subseteq S^*\). Hence the probability that \(s\) is accepted is at most the expected number of trials that run \(\mathcal{A}_{\mathrm{St}}\) on a subset of \(S^*\), which Proposition 3.10 with \(\Gamma=2\) bounds by the displayed quantity.

(c) \(G[S^*]\) is a tree and every vertex outside \(S^*\) is a claw, so Lemma 7.3 holds with equality, with \(\nu=2\) at the legs and \(\nu=0\) on the spine, and \(W_{S^*}=W_{F,\nu}\) for \(F=\Lambda_L[S^*]\). Root \(F\) at \(x_L\), with \(x_{i-1}\) a child of \(x_i\). By Lemma 7.4, the message of \(x_1\) is \[\mathbf u_y=\mathrm{Init}_y(0)\circ\mathrm{Init}_y(2)\circ\mathrm{Init}_y(2) =\Bigl(\tfrac y2\bigl(y+\tfrac14\bigr)^2,\ \tfrac{y^2}4+\tfrac y8,\ \tfrac1{64}\Bigr),\] and the message of \(x_i\) is \(\mathbf u_y\circ\mathrm{msg}_{x_{i-1}}\) for \(i\ge2\), which is linear in \(\mathrm{msg}_{x_{i-1}}\) with matrix \[\mathbf M_y=\begin{pmatrix}2u_1&u_1&u_1\\u_2+u_3&u_2&u_2/2\\0&u_3&u_3/2\end{pmatrix},\qquad (u_1,u_2,u_3)=\mathbf u_y .\] So \(W_{S^*}(y)=\mathrm{fin}(\mathbf M_y^{L-1}\mathbf u_y)\). The entries of \(\mathbf u_y\) and \(\mathbf M_y\) are polynomials in \(y\) with nonnegative coefficients, so they are nonnegative and nondecreasing in \(y\ge0\). Write \(f=(1,1,\frac12)\), so that \(\mathrm{fin}(P)=f^{\mathsf T}P\).

For the first bound, \(\mu_t(S^*)=0\) for \(t>3L\). For \(0\le t\le3L\) put \(y_t=t/(n-t)\in[0,\frac34]\). As in the proof of Lemma 3.9, \(\mu_t(S^*)\,y_t^{\,t}\le W_{S^*}(y_t)\) and \(\binom nt y_t^{\,t}\ge(1+y_t)^n/(n+1)\), so (also for \(t=0\), where \(y_t=0\)) \[\mu_t(S^*)\Big/\binom nt\le(n+1)\,W_{S^*}(y_t)/(1+y_t)^n .\] The certificate of Appendix F.3 covers \([0,\frac34]\) by finitely many intervals \([a,b]\) and gives for each a rational vector \(w>0\) and a rational \(\lambda>0\) with \(w^{\mathsf T}\mathbf M_b\le\lambda w^{\mathsf T}\) componentwise, \(c\,w^{\mathsf T}\mathbf u_b\le\lambda\) for \(c=\max_i f_i/w_i\), and \(1.71927^7\lambda\le(1+a)^7\). For \(y\in[a,b]\) this gives \[\begin{split} W_{S^*}(y)&\le f^{\mathsf T}\mathbf M_b^{L-1}\mathbf u_b\le c\,w^{\mathsf T}\mathbf M_b^{L-1}\mathbf u_b \le c\,\lambda^{L-1}w^{\mathsf T}\mathbf u_b\le\lambda^L\\ &\le(1+a)^{7L}\,1.71927^{-n}\le(1+y)^n\,1.71927^{-n}, \end{split}\] and with \(y=y_t\) the first bound of (c) follows. For the converse, at \(y^*=0.3453\) the same certificate gives a rational vector \(v>0\) and a rational \(\lambda'>0\) with \(\mathbf M_{y^*}v\ge\lambda'v\) componentwise and \((1+y^*)^7\le1.71928^7\lambda'\). With \(c'=\min_iu_i/v_i>0\) for \(\mathbf u_{y^*}=(u_1,u_2,u_3)\), we get \(W_{S^*}(y^*)\ge c'f^{\mathsf T}\mathbf M_{y^*}^{L-1}v\ge c'\lambda'^{L-1}f^{\mathsf T}v\), and by Lemma 3.9 \(\min_t\binom nt/\mu_t(S^*)\le(1+y^*)^n/W_{S^*}(y^*)\le C\cdot1.71928^n\) with \(C=\lambda'/(c'f^{\mathsf T}v)\). The numerical value in (c) and values for finite \(L\) are reported in Appendix F.3; they are not used in the proof.

For the consequence, suppose \(2^\psi<1.71927\). By (c), \(\max_t\mu_t(S^*)/\binom nt\le (n+1)\,1.71927^{-n}\) for all \(L\), so by (b) the probability that \(\gamma_{\mathrm{c}}\) is accepted is at most \(n^{O(1)}(2^\psi/1.71927)^n\), which tends to \(0\). As RS-\(\theta\) never accepts a value below \(\gamma_{\mathrm{c}}\), it then outputs a value above \(\gamma_{\mathrm{c}}\) with probability tending to \(1\).  ∎

The bound does not rely on \(\Lambda_L\) being a tree, on which Max Leaf Spanning Tree is easy: Proposition 7.9 holds for every graph \(G\), with \(L\ge2\), whose unique maximum-leaf spanning tree is \(\Lambda_L\): part (a) by the argument below, which replaces its proof, and parts (b) and (c) with their proofs verbatim. Indeed, a minimum connected dominating set \(D\) of \(G\) yields a spanning tree with at least \(n-|D|\) leaves (Lemma 5.1), which must be \(\Lambda_L\), so \(D\supseteq S^*\) and \(D=S^*\). Next, every vertex of \(S^*\) has degree at least \(3\) in \(\Lambda_L\) (a leg has its spine vertex and two claws; a spine vertex has two legs and, as \(L\ge2\), a spine neighbor), and the leaves of \(\Lambda_L\) are exactly the claws. Let \(e\) be an edge of \(G\) that is not in \(\Lambda_L\). If both ends of \(e\) lie in \(S^*\), the cycle of \(\Lambda_L+e\) runs through the subtree \(\Lambda_L[S^*]\); delete an edge \(f\ne e\) of it. Each end of \(f\) lies in \(S^*\) and loses at most one unit of degree, so keeps degree at least \(2\), and no other degree decreases, so the new spanning tree has exactly the claws as leaves. If \(e=cv\) joins a claw \(c\) with leg \(\ell\) to a vertex \(v\in S^*\setminus\{\ell\}\), delete the edge \(c\ell\), which lies on the cycle of \(\Lambda_L+e\): then \(c\) is still a leaf, \(\ell\) keeps degree \(2\), and again the leaves are exactly the claws. In both cases we obtain a spanning tree other than \(\Lambda_L\) with as many leaves, contrary to uniqueness. The remaining edges of \(G\) outside \(\Lambda_L\) join two claws; they change neither \(G[S^*]\) nor the neighbors in \(S^*\) of any claw. So \(G[S^*]=\Lambda_L[S^*]\) and each claw has exactly one neighbor in \(S^*\), which is what parts (b) and (c) use.

The extremal structure is neither the star of stars of Proposition 5.12, which is tight for Lemma 5.11 but has a smaller base than \(\Lambda_L\) under exponential moments, nor a path. With at most \(8\) pendant vertices per solution vertex, the base given by the generating function, computed as in Proposition 7.9(c), is at most \(1.5878\) for stars of stars and at most \(1.6972\) for paths (numerical; further families in Appendix F.3).

D.5 The closure bound of Algorithm RSC

This subsection proves Proposition 8.6, the lower bound on the closure generating function \(W^{\mathrm{cl}}_S\) of a minimum connected dominating set \(S\), in the three steps described in Section 8: the tree model (Appendix D.5.1), the ten-state recursion and the light owners (Appendices D.5.2 and D.5.3), and the antinorm certificate (Appendix D.5.4). The notation is that of Section 8.

Restatement of Proposition 8.6.

D.5.1 From the graph to a tree model

From now on \(G\) is a connected graph, \(S\) a connected dominating set of \(G\) with \(|S|=s\), \(F\) a spanning tree of \(G[S]\), and \(O=V\setminus S\) the set of outside vertices. For \(u\in O\) let \(N_S(u)=N(u)\cap S\), which is nonempty because \(S\) is dominating. An outside vertex \(u\) is private if \(|N_S(u)|=1\) and shared otherwise; a shared vertex has type \(\tau(u)=2\) if \(|N_S(u)|=2\) and \(\tau(u)=3\) if \(|N_S(u)|\ge3\). For \(X'\subseteq S\) and \(v\in S\) let \(k_F(v)=|N_F(v)\cap X'|\) be the number of sampled \(F\)-neighbors of \(v\) (the dependence on \(X'\) is suppressed), and let \[C_F(X')=\bigl\{v\in S\setminus X':\ k_F(v)\ge2\bigr\}\] be the candidates that the tree \(F\) certifies. For \(\Upsilon\subseteq C_F(X')\) let \[\pi_{X'}(\Upsilon)=\prod_{v\in\Upsilon}p_{k_F(v)}\prod_{v\in C_F(X')\setminus\Upsilon}q_{k_F(v)},\] the probability that a random subset of \(C_F(X')\) which contains each \(v\) independently with probability \(p_{k_F(v)}\) equals \(\Upsilon\). For \(X\subseteq S\) put \[\Psi_F(X)=\bigl|\{v\in S:\ N_F[v]\cap X=\emptyset\}\bigr|+\operatorname{comp}(F[X]),\] and for \(u\in O\) and \(X'\subseteq X\subseteq S\) put, with \(k(u)=|N(u)\cap X'|\), \[h_u(X',X)=2^{-\mathbf 1[N(u)\cap X=\emptyset]}\,q_{k(u)}^{\mathbf 1[k(u)\ge2]} .\] The factor \(h_u\) combines the factor \(\frac12\) that an outside vertex not dominated by the closed sample \(X\) contributes to \(2^{-|\mathrm{T}(J_X)|}\) with the probability \(q_{k(u)}\) that a candidate outside the solution with \(k(u)\) sampled neighbors is not put into \(\Upsilon\). Note that \(k(u)\le|N_S(u)|\), so \(k(u)\le2\) for a vertex of type \(2\).

We first record that more sampled vertices never mean more terminals. This is why adding candidates inside the solution never decreases \(2^{-|\mathrm{T}(J_X)|}\).

Lemma D.5.1 (Monotonicity of the number of terminals).  For \(X\subseteq Y\subseteq V\) we have \(|\mathrm{T}(J_Y)|\le|\mathrm{T}(J_X)|\).

Lean proof: rsc_mono (formalisation)

Proof. It suffices to add one vertex \(v\notin X\), and we use (13). If \(v\notin N[X]\), then \(|N[X\cup\{v\}]|\ge|N[X]|+1\), because \(v\) is new in the closed neighborhood, and \(\operatorname{comp}(G[X\cup\{v\}])\le\operatorname{comp}(G[X])+1\). If \(v\in N(X)\), then \(v\) is adjacent to at least one component of \(G[X]\), so \(\operatorname{comp}(G[X\cup\{v\}])\le\operatorname{comp}(G[X])\), and \(N[X\cup\{v\}]\supseteq N[X]\). In both cases \(n-|N[\cdot]|+\operatorname{comp}(G[\cdot])\) does not increase.  ∎

We shall use the following consequence of monotonicity several times. If \(f\) is a nonnegative function on the subsets of a finite set \(C\) that is nondecreasing under inclusion, and \(\mu_{\mathbf p}\) is the product distribution on the subsets of \(C\) that contains each \(v\in C\) independently with probability \(\mathbf p_v\), then \(\sum_\Upsilon\mu_{\mathbf p}(\Upsilon)f(\Upsilon)\) is nondecreasing in each \(\mathbf p_v\): writing the sum as \(\mathbf p_v\,\mathbb{E}[f(\Upsilon'\cup\{v\})]+(1-\mathbf p_v)\mathbb{E}[f(\Upsilon')]\) with \(\Upsilon'\) the random subset of \(C\setminus\{v\}\), its derivative in \(\mathbf p_v\) is \(\mathbb{E}[f(\Upsilon'\cup\{v\})-f(\Upsilon')]\ge0\). We call this the monotone coupling.

The next lemma replaces the graph by the tree \(F\). It keeps the candidates that \(F\) certifies, with the (smaller) closure probabilities that \(F\) certifies, drops the other candidates inside \(S\), and turns every candidate outside \(S\) into a factor \(q_{k(u)}\).

Lemma D.5.2 (Closure tree model).  For \(0\le p_2\le p_3\le1\) and \(y\ge0\), \[W^{\mathrm{cl}}_S(y)\ \ge\ \sum_{X'\subseteq S}\ \sum_{\Upsilon\subseteq C_F(X')} y^{|X'|}\,\pi_{X'}(\Upsilon)\,2^{-\Psi_F(X'\cup\Upsilon)}\prod_{u\in O}h_u(X',X'\cup\Upsilon).\]

Lean proof: rsc_model, monotone_coupling (formalisation)

Proof. Fix \(X'\subseteq S\) and let \(\Upsilon\) be the random closure of Definition 8.3. Split the candidate set \(C(X')\) into three disjoint parts: \(C^{\mathrm{out}}=C(X')\cap O\), the set \(C_F(X')\), and \(C^{\mathrm{rest}}=(C(X')\cap S)\setminus C_F(X')\). These parts cover \(C(X')\) because \(C_F(X')\subseteq C(X')\cap S\): a vertex \(v\in S\setminus X'\) with two \(F\)-neighbors in \(X'\) has two \(G\)-neighbors in \(X'\), as \(F\subseteq G\). The sets \(\Upsilon\cap C^{\mathrm{out}}\), \(\Upsilon'=\Upsilon\cap C_F(X')\) and \(\Upsilon''=\Upsilon\cap C^{\mathrm{rest}}\) are independent, and the event \(\Upsilon\subseteq S\) is the event \(\Upsilon\cap C^{\mathrm{out}}=\emptyset\), which has probability \(\prod_{u\in C^{\mathrm{out}}}q_{|N(u)\cap X'|}\). On this event \(X'\cup\Upsilon\supseteq X'\cup\Upsilon'\), so \(2^{-|\mathrm{T}(J_{X'\cup\Upsilon})|}\ge2^{-|\mathrm{T}(J_{X'\cup\Upsilon'})|}\) by Lemma D.5.1. The set \(\Upsilon'\) contains each \(v\in C_F(X')\) independently with probability \(p_{|N(v)\cap X'|}\ge p_{k_F(v)}\), because \(|N(v)\cap X'|\ge k_F(v)\) and \(p_2\le p_3\), and \(\Upsilon'\mapsto2^{-|\mathrm{T}(J_{X'\cup\Upsilon'})|}\) is nondecreasing by Lemma D.5.1; so the monotone coupling gives \[\mathbb{E}_\Upsilon\Bigl[\mathbf 1[\Upsilon\subseteq S]\,2^{-|\mathrm{T}(J_{X'\cup\Upsilon})|}\Bigr] \ \ge\ \prod_{u\in C^{\mathrm{out}}}q_{|N(u)\cap X'|}\ \sum_{\Upsilon'\subseteq C_F(X')}\pi_{X'}(\Upsilon')\,2^{-|\mathrm{T}(J_{X'\cup\Upsilon'})|} .\] A vertex \(u\in O\) is not in \(X'\subseteq S\), so \(u\in C^{\mathrm{out}}\) if and only if \(k(u)\ge2\), and the product over \(C^{\mathrm{out}}\) equals \(\prod_{u\in O}q_{k(u)}^{\mathbf 1[k(u)\ge2]}\).

Now fix \(X\subseteq S\) (we shall take \(X=X'\cup\Upsilon'\)). By (13), \(|\mathrm{T}(J_X)|=|S\setminus N[X]|+|O\setminus N[X]|+\operatorname{comp}(G[X])\). First, if \(v\in S\) is not in \(N[X]\), then \(N_G[v]\cap X=\emptyset\) and hence \(N_F[v]\cap X=\emptyset\), since \(N_F[v]\subseteq N_G[v]\). Second, a vertex \(u\in O\) is not in \(X\), so \(u\notin N[X]\) if and only if \(N(u)\cap X=\emptyset\). Third, \(F[X]\) is a spanning subgraph of \(G[X]\), because \(X\subseteq S=V(F)\), so \(\operatorname{comp}(G[X])\le\operatorname{comp}(F[X])\); for \(X=\emptyset\) both sides are \(0\). Together, \[|\mathrm{T}(J_X)|\ \le\ \Psi_F(X)+\sum_{u\in O}\mathbf 1[N(u)\cap X=\emptyset] .\] Inserting this with \(X=X'\cup\Upsilon'\) into the previous display, multiplying by \(y^{|X'|}\) and summing over \(X'\subseteq S\) gives the claim.  ∎

The factors \(h_u\) couple the outside vertices with all of their neighbors in \(S\). To obtain a recursion along \(F\) we assign every outside vertex to one of its neighbors in \(S\), its owner.

Definition D.5.3 (Owners).  An owner map is a map \(o:O\to S\) with \(o(u)\in N_S(u)\) for all \(u\in O\). For \(v\in S\) let \(\nu_v=|o^{-1}(v)|\) be the number of vertices that \(v\) owns, so \(\sum_{v\in S}\nu_v=n-s\). For \(u\in O\) and \(X\subseteq S\) let \(\bar h_u(X)=2^{-\mathbf 1[o(u)\notin X]}\).

The following pointwise inequality compares \(h_u\) with the owner factor \(\bar h_u\); it is the only place where the values of \(h_u\) are examined. Put \[\varepsilon_2=-\ln q_2,\qquad \varepsilon_3=-\ln q_3,\qquad \delta_2=\ln2,\qquad \delta_3=\ln(2q_3),\] so \(0\le\varepsilon_2\le\varepsilon_3\) and \(0\le\delta_3\le\delta_2\) (recall \(q_2\ge q_3>\frac12\)). For \(u\in O\) let \(N'_u=N_S(u)\setminus\{o(u)\}\), and for \(X'\subseteq S\) let \(\mathbf 1_u=\mathbf 1[N'_u\cap X'\ne\emptyset]\) indicate that some other neighbor of \(u\) in \(S\) is sampled.

Lemma D.5.4 (Pointwise bound).  Let \(0<p_2\le p_3<\frac12\), let \(o\) be an owner map, let \(u\in O\) have the owner \(o=o(u)\), and let \(X'\subseteq X\subseteq S\). Then \(h_u(X',X)\ge\bar h_u(X)\) whenever \(o\notin X\), and:

(i)  if \(u\) is private, then \(h_u(X',X)\ge\bar h_u(X)\);

(ii)  if \(u\) is shared of type \(\tau\), then \(h_u(X',X)\ge\bar h_u(X)\,q_\tau^{\mathbf 1[o\in X]}\);

(iii)  if \(u\) is shared of type \(\tau\), \(o\notin X\) and \(N'_u\cap X\ne\emptyset\), then \(h_u(X',X)\ge e^{\delta_\tau}\,\bar h_u(X)\);

(iv)  if \(u\) is shared of type \(3\), then \(\ln\bigl(h_u(X',X)/\bar h_u(X)\bigr)\ge\mathbf 1_u\bigl(\delta_3\,\mathbf 1[o\notin X]-\varepsilon_3\,\mathbf 1[o\in X]\bigr)\);

(v)  if \(u\) is shared of type \(2\), then \(\ln\bigl(h_u(X',X)/\bar h_u(X)\bigr)\ge\mathbf 1_u\bigl(\delta_2\,\mathbf 1[o\notin X]-\varepsilon_2\,\mathbf 1[o\in X']\bigr)\).

Lean proof: rsc_pointwise (formalisation)

Proof. Suppose first that \(o\notin X\), so \(\bar h_u(X)=\frac12\) and \(o\notin X'\). If \(N'_u\cap X\ne\emptyset\), then \(u\) is dominated and \(h_u(X',X)=q_{k(u)}^{\mathbf 1[k(u)\ge2]}\ge q_3\); so \(h_u/\bar h_u\ge2q_3=e^{\delta_3}\), and if moreover \(\tau(u)=2\), then \(k(u)\le|N'_u|=1\) gives \(h_u=1\) and \(h_u/\bar h_u=2=e^{\delta_2}\). This proves (iii), and it covers the case \(\mathbf 1_u=1\) of (iv) and (v), since \(X'\subseteq X\). If \(N'_u\cap X=\emptyset\), then \(N(u)\cap X'=\emptyset\), so \(h_u(X',X)=2^{-\mathbf 1[N(u)\cap X=\emptyset]}\ge\frac12=\bar h_u(X)\). In all cases \(h_u\ge\bar h_u\) when \(o\notin X\).

Suppose now that \(o\in X\), so \(\bar h_u(X)=1\) and \(u\) is dominated; then \(h_u(X',X)=q_{k(u)}^{\mathbf 1[k(u)\ge2]}\). If \(\mathbf 1_u=0\), then \(N(u)\cap X'\subseteq\{o\}\), so \(h_u=1\). If \(\mathbf 1_u=1\), then \(h_u\ge q_3\), which gives (iv); if moreover \(\tau(u)=2\), then \(k(u)\le2\) and \(k(u)=2\) only if \(o\in X'\), so \(h_u\ge q_2^{\mathbf 1[o\in X']}\), which gives (v); and in both cases \(h_u\ge q_{\tau(u)}\), which gives (ii), as \(q_\tau\le1\). For (i), a private \(u\) has \(N'_u=\emptyset\), so \(k(u)\le1\) and \(h_u=1\) when \(o\in X\).  ∎

Replacing \(h_u\) by \(\bar h_u q_\tau^{\mathbf 1[o(u)\in X]}\) for every shared vertex would give a model of product form along the tree. We call the factor \(q_\tau^{\mathbf 1[o(u)\in X]}\) an exposure of the owner of \(u\), of type \(\tau=\tau(u)\). Exposing every shared vertex loses too much (for \(p_2=p_3=1/20\), about \(2^{0.012n}\) in the running time on the uniform \((2,2)\)-lobster \(\Lambda_L\) of Proposition 7.9, numerically), because the exposure applies the factor \(q_\tau\) to a shared vertex whenever its owner is in \(X\), although \(h_u<1\) requires at least two sampled neighbors. For some shared vertices we therefore keep the exact factor \(h_u\) and bound its effect by a correction term, using Jensen’s inequality. For the others, (iii) recovers part of the loss: when the owner is outside the closed sample, an exposed vertex is usually dominated by one of its other neighbors, and then \(h_u\) is \(e^{\delta_\tau}\) times the owner factor \(\frac12\). The tree model records this as a credit \(\varpi_\tau\ge1\) on the factor of an owner outside the closed sample.

Definition D.5.5 (Tree model).  Let \(o\) be an owner map, let the shared vertices be partitioned into a set \(O^{\mathrm{co}}\) of corrected vertices and a set \(O^{\mathrm{ex}}\) of exposed vertices, and let \(\varpi_2,\varpi_3\ge1\) be real numbers, the credits. For \(v\in S\) let \(\nu_v\) be as in Definition D.5.3, let \(c_{v,\tau}=|\{u\in o^{-1}(v)\cap O^{\mathrm{ex}}:\tau(u)=\tau\}|\) for \(\tau\in\{2,3\}\) be the numbers of exposed vertices of each type that \(v\) owns, and \(c_v=c_{v,2}+c_{v,3}\le\nu_v\). For \(X\subseteq S\) let \[\hat h_v(X)= \begin{cases}q_2^{c_{v,2}}q_3^{c_{v,3}}&\text{if }v\in X,\\[2pt] 2^{-\nu_v}\varpi_2^{c_{v,2}}\varpi_3^{c_{v,3}}&\text{if }v\notin X, \end{cases}\] and for \(X'\subseteq S\) and \(\Upsilon\subseteq C_F(X')\) let \[\mathrm{wt}(X',\Upsilon)=y^{|X'|}\,\pi_{X'}(\Upsilon)\,2^{-\Psi_F(X'\cup\Upsilon)}\prod_{v\in S}\hat h_v(X'\cup\Upsilon),\qquad W^{\mathrm{tm}}=\sum_{X'\subseteq S}\ \sum_{\Upsilon\subseteq C_F(X')}\mathrm{wt}(X',\Upsilon).\] Let \(\Pr_{\mathrm{tm}}\) be the probability distribution on the pairs \((X',\Upsilon)\) with \(\Pr_{\mathrm{tm}}(X',\Upsilon)=\mathrm{wt}(X',\Upsilon)/W^{\mathrm{tm}}\), and write \(X=X'\cup\Upsilon\) for the closed sample under \(\Pr_{\mathrm{tm}}\); we write \(\mathbb{E}_{\mathrm{tm}}\) for the expectation under \(\Pr_{\mathrm{tm}}\).

The value \(W^{\mathrm{tm}}\) depends only on \(F\), \(y\), \(p_2\), \(p_3\), \(\varpi_2\), \(\varpi_3\) and the numbers \(\nu_v\), \(c_{v,2}\) and \(c_{v,3}\); we write \(W^{\mathrm{tm}}(F,\nu,c)\) when this matters. It is positive: the pair \((\emptyset,\emptyset)\) has weight \(2^{-s}\prod_v2^{-\nu_v}\varpi_2^{c_{v,2}}\varpi_3^{c_{v,3}}>0\), since \(C_F(\emptyset)=\emptyset\). For \(u\in O\) put \(\tilde h_u=h_u(X',X)/\bar h_u(X)\), a random variable under \(\Pr_{\mathrm{tm}}\); it is positive when \(p_3<1\). For an exposed \(u\) of type \(\tau\) with owner \(o\) put also \[\tilde h^{\mathrm{ex}}_u=\frac{h_u(X',X)}{\bar h_u(X)\,q_\tau^{\mathbf 1[o\in X]}\,\varpi_\tau^{\mathbf 1[o\notin X]}} .\]

Lemma D.5.6 (Exposures, credits and the Jensen correction).  Let \(0<p_2\le p_3<\frac12\), \(y\ge0\) and \(\varpi_2,\varpi_3\ge1\). In the setting of Definition D.5.5, \[W^{\mathrm{cl}}_S(y)\ \ge\ W^{\mathrm{tm}}\exp\Bigl(\sum_{u\in O^{\mathrm{co}}}\mathbb{E}_{\mathrm{tm}}[\ln\tilde h_u] +\sum_{u\in O^{\mathrm{ex}}}\mathbb{E}_{\mathrm{tm}}[\ln\tilde h^{\mathrm{ex}}_u]\Bigr).\] In particular, if all these expectations are nonnegative, then \(W^{\mathrm{cl}}_S(y)\ge W^{\mathrm{tm}}\).

Lean proof: rsc_jensen, EQ3, EQ2, EQex, rsc_jensen_nonneg (formalisation)

Proof. Let \(X'\subseteq S\), \(\Upsilon\subseteq C_F(X')\) and \(X=X'\cup\Upsilon\). Grouping the outside vertices by their owners, \[\prod_{u\in O}\bar h_u(X)\prod_{u\in O^{\mathrm{ex}}}q_{\tau(u)}^{\mathbf 1[o(u)\in X]}\varpi_{\tau(u)}^{\mathbf 1[o(u)\notin X]}=\prod_{v\in S}\hat h_v(X),\] because the vertices owned by \(v\in X\) contribute \(1\) each, except the exposed ones, which contribute \(q_{\tau(u)}\), and the \(\nu_v\) vertices owned by \(v\notin X\) contribute \(\frac12\) each, the exposed ones with the extra factor \(\varpi_{\tau(u)}\). Hence the summand for \((X',\Upsilon)\) on the right-hand side of Lemma D.5.2 equals \[\mathrm{wt}(X',\Upsilon)\prod_{u\in O^{\mathrm{ex}}}\tilde h^{\mathrm{ex}}_u\ \prod_{u\in O^{\mathrm{co}}}\tilde h_u \prod_{u\text{ private}}\frac{h_u(X',X)}{\bar h_u(X)} \ \ge\ \mathrm{wt}(X',\Upsilon)\prod_{u\in O^{\mathrm{ex}}}\tilde h^{\mathrm{ex}}_u\prod_{u\in O^{\mathrm{co}}}\tilde h_u ,\] because the private factors are at least \(1\) by Lemma D.5.4(i). Summing over \((X',\Upsilon)\) and using Lemma D.5.2, \[W^{\mathrm{cl}}_S(y)\ \ge\ W^{\mathrm{tm}}\,\mathbb{E}_{\mathrm{tm}}\Bigl[\exp\Bigl(\sum_{u\in O^{\mathrm{co}}}\ln\tilde h_u+\sum_{u\in O^{\mathrm{ex}}}\ln\tilde h^{\mathrm{ex}}_u\Bigr)\Bigr] \ \ge\ W^{\mathrm{tm}}\exp\Bigl(\mathbb{E}_{\mathrm{tm}}\Bigl[\sum_{u\in O^{\mathrm{co}}}\ln\tilde h_u+\sum_{u\in O^{\mathrm{ex}}}\ln\tilde h^{\mathrm{ex}}_u\Bigr]\Bigr),\] where the last step is Jensen’s inequality for the convex function \(\exp\); all the random variables are positive because \(q_3>0\).  ∎

Taking expectations in Lemma D.5.4(iv) and (v), for every corrected \(u\), \[\begin{align*}\mathbb{E}_{\mathrm{tm}}[\ln\tilde h_u]&\ \ge\ \delta_3\,\Pr_{\mathrm{tm}}\bigl(\mathbf 1_u=1,\ o(u)\notin X\bigr)-\varepsilon_3\,\Pr_{\mathrm{tm}}\bigl(\mathbf 1_u=1,\ o(u)\in X\bigr) &&\text{if }\tau(u)=3,\tag{26}\\ \mathbb{E}_{\mathrm{tm}}[\ln\tilde h_u]&\ \ge\ \delta_2\,\Pr_{\mathrm{tm}}\bigl(\mathbf 1_u=1,\ o(u)\notin X\bigr)-\varepsilon_2\,\Pr_{\mathrm{tm}}\bigl(\mathbf 1_u=1,\ o(u)\in X'\bigr) &&\text{if }\tau(u)=2.\tag{27}\end{align*}\] For an exposed \(u\) of type \(\tau\) with owner \(o\), Lemma D.5.4(ii) gives \(\ln\tilde h^{\mathrm{ex}}_u\ge0\) on the event \(o\in X\), and on the event \(o\notin X\) we have \(\ln\tilde h^{\mathrm{ex}}_u=\ln\tilde h_u-\ln\varpi_\tau\) with \(\ln\tilde h_u\ge\delta_\tau\mathbf 1[N'_u\cap X\ne\emptyset]\) by (iii) and the first assertion of the lemma; hence \[\tag{28} \mathbb{E}_{\mathrm{tm}}[\ln\tilde h^{\mathrm{ex}}_u]\ \ge\ \Pr_{\mathrm{tm}}(o\notin X)\Bigl(\delta_\tau\Pr_{\mathrm{tm}}\bigl(N'_u\cap X\ne\emptyset\bigm|o\notin X\bigr)-\ln\varpi_\tau\Bigr)\] Every event of the form \(U\cap X=\emptyset\) has positive probability under \(\Pr_{\mathrm{tm}}\), because the pair \((\emptyset,\emptyset)\) has positive weight; in particular \(\Pr_{\mathrm{tm}}(o\notin X)>0\), so the conditional probability is defined. Appendix D.5.3 shows that the right-hand sides of (26) and (27) are nonnegative when the owner \(o(u)\) is light for the type of \(u\) (Definition D.5.10): it owns few vertices and has small degree in \(F\). It also bounds the conditional probability in (28) from below, which yields the admissible credits. To compute the probabilities we need a recursion for the tree model.

D.5.2 The ten-state recursion

As in Lemma 7.4, the tree model is a sum over configurations of a product of local factors, which can be evaluated from the leaves of \(F\) upwards. The weight \(\mathrm{wt}(X',\Upsilon)\) factorizes over the vertices and edges of \(F\). Since \(F[X]\) is a forest, \(\operatorname{comp}(F[X])=|X|-|E(F[X])|\), so \[2^{-\Psi_F(X)}=\prod_{v\in X}\tfrac12\ \prod_{vw\in E(F[X])}2\ \prod_{v\in S:\,N_F[v]\cap X=\emptyset}\tfrac12 ;\] and \(\pi_{X'}(\Upsilon)\) is the product, over the vertices \(v\in S\setminus X'\), of \(p_{k_F(v)}\) if \(v\in\Upsilon\), of \(q_{k_F(v)}\) if \(v\in C_F(X')\setminus\Upsilon\), and of \(1\) otherwise. The factor of a vertex depends on its own status and on how many of its \(F\)-neighbors lie in \(X'\) and in \(X\); since \(p_k\) and \(q_k\) depend on \(k\) only through \(\min(k,3)\), a vertex has to count its sampled neighbors up to three. Write \[\lambda=p_3/p_2\ \ge1 .\] A message records the status of a vertex together with the information about its children that is still needed once the status of its parent is known.

States.

Root \(F\) at a vertex \(r\). Fix a vertex \(v\) and a configuration on the subtree formed by \(v\) and some of its children’s subtrees. The state of \(v\) is one of the following ten; “child” refers to the children included so far, and “sampled” means “in \(X'\).”

• \(\mathsf A\): \(v\in X'\).

• \(\mathsf C_0\), \(\mathsf C_1\), \(\mathsf C_2\), \(\mathsf C_3\): \(v\in\Upsilon\), and \(v\) has \(0\), \(1\), \(2\), or at least \(3\) sampled children. When the configuration is completed, \(v\in\Upsilon\) is allowed only if \(v\in C_F(X')\), that is, if the number of sampled children plus \(\mathbf 1[\text{parent}\in X']\) is at least \(2\).

• \(\mathsf O_{00}\): \(v\notin X\), and no child of \(v\) is in \(X\).

• \(\mathsf O_{01}\): \(v\notin X\), no child of \(v\) is sampled, and some child is in \(\Upsilon\).

• \(\mathsf O_1\), \(\mathsf O_2\), \(\mathsf O_3\): \(v\notin X\), and exactly one, exactly two, or at least three children of \(v\) are sampled.

A message is a vector \(P\in\mathbb{R}^{10}\) indexed by these states, in this order; we write its coordinates as \(P=(A,C_0,C_1,C_2,C_3,O_{00},O_{01},O_1,O_2,O_3)\) for a parent and \(Q=(a,c_0,c_1,c_2,c_3,o_{00},o_{01},o_1,o_2,o_3)\) for a child. (These letters denote numbers; the states \(\mathsf C_j\) are those of the vertices in the closure \(\Upsilon\).) The weight of a vertex in \(\Upsilon\) is carried as \(\frac{p_2}2\), and the factor \(\lambda\) that turns \(p_2\) into \(p_3\) is applied when the vertex turns out to have at least three sampled neighbors.

Merge.

For a message \(Q\) of a child let \[\begin{gathered} \chi_{\mathsf S}(Q)=2a+2c_1+2\lambda(c_2+c_3)+o_{00}+o_{01}+q_2o_1+q_3(o_2+o_3),\\ \chi^{\mathrm s}_{\mathsf C}(Q)=2a,\qquad \chi^{\mathrm u}_{\mathsf C}(Q)=2c_2+2\lambda c_3+o_{00}+o_{01}+o_1+q_2o_2+q_3o_3,\\ \chi^{\mathrm s}_{\mathsf O}(Q)=a,\qquad \chi^{\mathrm x}_{\mathsf O}(Q)=c_2+\lambda c_3,\qquad \chi^{\mathrm u}_{\mathsf O}(Q)=\tfrac12o_{00}+o_{01}+o_1+q_2o_2+q_3o_3, \end{gathered}\] and \(\chi_{\mathsf O}(Q)=\chi^{\mathrm s}_{\mathsf O}(Q)+\chi^{\mathrm x}_{\mathsf O}(Q)+\chi^{\mathrm u}_{\mathsf O}(Q)=a+c_2+\lambda c_3+\tfrac12o_{00}+o_{01}+o_1+q_2o_2+q_3o_3\). These are the total weights that the child contributes when the parent is sampled (\(\chi_{\mathsf S}\)); when the parent is in \(\Upsilon\), split by whether the child is sampled (\(\chi^{\mathrm s}_{\mathsf C}\)) or not (\(\chi^{\mathrm u}_{\mathsf C}\)); and when the parent is not in \(X\), split by whether the child is sampled (\(\chi^{\mathrm s}_{\mathsf O}\)), is in \(\Upsilon\) (\(\chi^{\mathrm x}_{\mathsf O}\)), or neither (\(\chi^{\mathrm u}_{\mathsf O}\)). Writing \(\chi^{\mathrm s}_{\mathsf C}\) for \(\chi^{\mathrm s}_{\mathsf C}(Q)\) and so on, define \[P\circ Q= \begin{pmatrix} A\,\chi_{\mathsf S}(Q)\\ C_0\chi^{\mathrm u}_{\mathsf C}\\ C_1\chi^{\mathrm u}_{\mathsf C}+C_0\chi^{\mathrm s}_{\mathsf C}\\ C_2\chi^{\mathrm u}_{\mathsf C}+C_1\chi^{\mathrm s}_{\mathsf C}\\ C_3(\chi^{\mathrm u}_{\mathsf C}+\chi^{\mathrm s}_{\mathsf C})+C_2\chi^{\mathrm s}_{\mathsf C}\\ O_{00}\chi^{\mathrm u}_{\mathsf O}\\ O_{01}(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O})+O_{00}\chi^{\mathrm x}_{\mathsf O}\\ O_1(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O})+(O_{00}+O_{01})\chi^{\mathrm s}_{\mathsf O}\\ O_2(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O})+O_1\chi^{\mathrm s}_{\mathsf O}\\ O_3(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O}+\chi^{\mathrm s}_{\mathsf O})+O_2\chi^{\mathrm s}_{\mathsf O} \end{pmatrix}^{\!\top},\] and \[\mathrm{fin}(P)=A+C_2+\lambda C_3+\tfrac12O_{00}+O_{01}+O_1+q_2O_2+q_3O_3 .\] The operation \(\circ\) is bilinear and maps nonnegative messages to nonnegative messages.

Initial messages and forcing.

For integers \(\nu\ge0\) and \(c_2,c_3\ge0\) let \[\mathrm{Init}(\nu;c_2,c_3)=\bigl(q_2^{c_2}q_3^{c_3}\tfrac y2,\ q_2^{c_2}q_3^{c_3}\tfrac{p_2}2,\ 0,\ 0,\ 0,\ 2^{-\nu}\varpi_2^{c_2}\varpi_3^{c_3},\ 0,\ 0,\ 0,\ 0\bigr).\] A forcing is a map \(f\) from a subset of \(S\) to \(\{\mathsf{in},\mathsf{out},\mathsf{excl}\}\); it restricts the configurations to those with \(v\in X'\) if \(f(v)=\mathsf{in}\), with \(v\notin X'\) if \(f(v)=\mathsf{out}\), and with \(v\notin X\) if \(f(v)=\mathsf{excl}\) (the vertex is excluded; it may be a candidate, but it is not put into \(\Upsilon\)). For a forced vertex we use the forced initial message: \(\mathrm{Init}(\nu;c_2,c_3)\) with all coordinates except the first set to \(0\) if \(f(v)=\mathsf{in}\), with the first coordinate set to \(0\) if \(f(v)=\mathsf{out}\), and with the first two coordinates set to \(0\) if \(f(v)=\mathsf{excl}\). Note that \(f(v)=\mathsf{out}\) still allows \(v\in\Upsilon\).

Definition D.5.7 (Messages).  Let \(F\) be rooted at \(r\), let \(\nu,c_2,c_3:S\to\mathbb Z_{\ge0}\), let \(0<p_2\le p_3<1\), \(y\ge0\), \(\varpi_2,\varpi_3\ge1\), and let \(f\) be a forcing. For every \(v\in S\) fix an order \(w_1,\dots,w_d\) of its children. The messages \(M_v\), \(v\in S\), are defined from the leaves upwards by \[M_v=\Bigl(\cdots\bigl(\mathrm{Init}_f(\nu_v;c_{v,2},c_{v,3})\circ M_{w_1}\bigr)\circ\cdots\Bigr)\circ M_{w_d},\] where \(\mathrm{Init}_f\) is the forced initial message if \(v\) is forced and \(\mathrm{Init}\) otherwise.

Lemma D.5.8 (Ten-state recursion).  Let \(M_v\), \(v\in S\), be the messages of Definition D.5.7, for any orders of the children. Then every \(M_v\) is nonnegative, and with the weights \(\mathrm{wt}\) of Definition D.5.5 \[\begin{aligned} A_r&=\sum_{(X',\Upsilon)\models f,\ r\in X'}\mathrm{wt}(X',\Upsilon),\\ (C_2)_r+\lambda(C_3)_r&=\sum_{(X',\Upsilon)\models f,\ r\in\Upsilon}\mathrm{wt}(X',\Upsilon),\\ \tfrac12(O_{00})_r+(O_{01})_r+(O_1)_r+q_2(O_2)_r+q_3(O_3)_r&=\sum_{(X',\Upsilon)\models f,\ r\notin X}\mathrm{wt}(X',\Upsilon), \end{aligned}\] where \(M_r=(A_r,\dots,(O_3)_r)\) and \((X',\Upsilon)\models f\) ranges over the pairs with \(X'\subseteq S\) and \(\Upsilon\subseteq C_F(X')\) that respect the forcing. In particular \(\mathrm{fin}(M_r)\) is the sum of \(\mathrm{wt}\) over these pairs, and \(\mathrm{fin}(M_r)=W^{\mathrm{tm}}\) for the empty forcing.

Lean proof: rsc_recursion, rsc_recursion_fin (formalisation)

Proof. The proof assigns to every partial configuration on a subtree a weight that collects those factors of \(\mathrm{wt}(X',\Upsilon)\) which are already determined inside the subtree, and it shows by induction along the merges that each message sums these weights state by state. At the root nothing remains undetermined, and the displayed sums follow.

For a vertex \(v\) and \(0\le i\le d\) let \(F_v^i\) be the subtree formed by \(v\) and the subtrees of \(w_1,\dots,w_i\); write \(F_v\) for \(F_v^d\). A partial configuration on a vertex set \(U\) is a pair \((X'_U,\Upsilon_U)\) of disjoint subsets of \(U\) respecting the forcing. For a partial configuration on \(V(F_v^i)\), define its partial weight as the product of the following factors, where \(k\) denotes the number of sampled \(F\)-neighbors of the vertex in question:

• for every vertex \(x\ne v\) of \(F_v^i\) (whose \(F\)-neighbors all lie in \(F_v^i\), since the parent of \(x\) is in \(F_v^i\)): the complete vertex factor, namely \(\frac y2q_2^{c_{x,2}}q_3^{c_{x,3}}\) if \(x\in X'\); \(\frac{p_k}2q_2^{c_{x,2}}q_3^{c_{x,3}}\) if \(x\in\Upsilon\) and \(k\ge2\), and \(0\) if \(x\in\Upsilon\) and \(k\le1\); if \(x\notin X'\cup\Upsilon\), the product of \(2^{-\nu_x}\varpi_2^{c_{x,2}}\varpi_3^{c_{x,3}}\), of \(q_k\) if \(k\ge2\), and of \(\frac12\) if no \(F\)-neighbor of \(x\) is in \(X'\cup\Upsilon\);

• for \(v\) itself, the partial vertex factor: \(\frac y2q_2^{c_{v,2}}q_3^{c_{v,3}}\) if \(v\in X'\), \(\frac{p_2}2q_2^{c_{v,2}}q_3^{c_{v,3}}\) if \(v\in\Upsilon\), and \(2^{-\nu_v}\varpi_2^{c_{v,2}}\varpi_3^{c_{v,3}}\) if \(v\notin X'\cup\Upsilon\);

• a factor \(2\) for every edge of \(F_v^i\) with both ends in \(X'\cup\Upsilon\).

By the factorization displayed before the definition of the states, the product of the complete vertex factors of all vertices of \(S\) and of the edge factors is \(\mathrm{wt}(X',\Upsilon)\) for \(\Upsilon\subseteq C_F(X')\), and \(0\) if \(\Upsilon\) contains a vertex with fewer than two sampled \(F\)-neighbors. (For the vertex factor of \(x\in\Upsilon\), the requirement \(x\in C_F(X')\) is exactly that \(x\) has at least two sampled \(F\)-neighbors; the factor \(\frac12\) per vertex of \(X\), the factor \(p_k\) or \(q_k\) of \(\pi_{X'}\), and the factor \(\hat h_x\) are all included.) Let \(M_v^i\in\mathbb{R}^{10}\) be the vector whose coordinate for a state is the sum of the partial weights of the partial configurations on \(V(F_v^i)\) in which \(v\) is in that state. We show by induction that \(M_v^i\) is the message obtained after merging \(w_1,\dots,w_i\); for \(i=d\) this is \(M_v\), which is then nonnegative.

For \(i=0\), \(F_v^0=\{v\}\); the configurations are \(v\in X'\) (state \(\mathsf A\), weight \(\frac y2q_2^{c_{v,2}}q_3^{c_{v,3}}\)), \(v\in\Upsilon\) (state \(\mathsf C_0\), weight \(\frac{p_2}2q_2^{c_{v,2}}q_3^{c_{v,3}}\)) and \(v\notin X'\cup\Upsilon\) (state \(\mathsf O_{00}\), weight \(2^{-\nu_v}\varpi_2^{c_{v,2}}\varpi_3^{c_{v,3}}\)), restricted by the forcing. This is \(\mathrm{Init}_f(\nu_v;c_{v,2},c_{v,3})\).

For the step, let \(w=w_i\) and \(Q=M_w\), which by induction collects the partial configurations of \(F_w\) with \(w\) as the special vertex. A partial configuration on \(V(F_v^i)\) is a pair of partial configurations on \(V(F_v^{i-1})\) and \(V(F_w)\). Its partial weight is the product of their partial weights, of the factor that completes the vertex factor of \(w\) (now that the status of its parent \(v\) is known), and of the factor \(2\) if \(v,w\in X\); the partial vertex factor of \(v\) does not change. We list, for each status of \(v\) and each state of \(w\), the completing factor, the edge factor, and the new state of \(v\). If \(w\) is in state \(\mathsf C_j\) or \(\mathsf O_j\), it has \(j\) sampled children (counted up to \(3\), with \(j=0\) for \(\mathsf O_{00}\) and \(\mathsf O_{01}\)).

• \(v\in X'\). If \(w\) is in state \(\mathsf A\), the edge gives \(2\). If \(w\) is in state \(\mathsf C_j\), it has \(k=j+1\) sampled neighbors, so it is a valid candidate if and only if \(j\ge1\), with the completing factor \(p_k/p_2\), which is \(1\) for \(j=1\) and \(\lambda\) for \(j\ge2\); the edge gives \(2\). If \(w\) is in an \(\mathsf O\)-state, it is dominated by \(v\), and it has \(k=j+1\) sampled neighbors, so it is a candidate not put into \(\Upsilon\), with factor \(q_k\), if and only if \(j\ge1\): \(q_2\) in \(\mathsf O_1\) and \(q_3\) in \(\mathsf O_2\) and \(\mathsf O_3\). The state of \(v\) remains \(\mathsf A\). The total is \(A\,\chi_{\mathsf S}(Q)\).

• \(v\in\Upsilon\). A sampled child contributes \(2a=\chi^{\mathrm s}_{\mathsf C}\) (edge factor \(2\)) and raises the number of sampled children of \(v\) by one. A child in \(\Upsilon\) has an unsampled parent, so it has \(k=j\) sampled neighbors and is valid only in the states \(\mathsf C_2\) (factor \(1\)) and \(\mathsf C_3\) (factor \(\lambda\)), with edge factor \(2\); a child not in \(X\) is dominated by \(v\), has \(k=j\) sampled neighbors, and so gets the factor \(q_2\) in state \(\mathsf O_2\) and \(q_3\) in state \(\mathsf O_3\). These cases contribute \(\chi^{\mathrm u}_{\mathsf C}\) and leave the count of \(v\) unchanged. This gives the second to fifth coordinates of \(P\circ Q\), with the count capped at \(3\).

• \(v\notin X\). A sampled child contributes \(a=\chi^{\mathrm s}_{\mathsf O}\) and moves \(v\) from \(\mathsf O_{00}\) or \(\mathsf O_{01}\) to \(\mathsf O_1\), from \(\mathsf O_1\) to \(\mathsf O_2\), and from \(\mathsf O_2\) or \(\mathsf O_3\) to \(\mathsf O_3\). A child in \(\Upsilon\) is valid only in the states \(\mathsf C_2\) and \(\mathsf C_3\), contributes \(c_2+\lambda c_3=\chi^{\mathrm x}_{\mathsf O}\), dominates \(v\), and moves \(v\) from \(\mathsf O_{00}\) to \(\mathsf O_{01}\). A child not in \(X\) has no dominating parent: it gets \(\frac12\) in state \(\mathsf O_{00}\), \(q_2\) in state \(\mathsf O_2\) and \(q_3\) in state \(\mathsf O_3\); this contributes \(\chi^{\mathrm u}_{\mathsf O}\) and leaves the state of \(v\) unchanged. This gives the last five coordinates of \(P\circ Q\).

In all cases no other vertex factor changes, because only \(v\) and \(w\) gain a neighbor with known status. So \(M_v^i=M_v^{i-1}\circ M_w\).

At the root, the configurations on \(V(F_r)=S\) are complete once the partial vertex factor of \(r\) is completed, and \(r\) has no parent. In state \(\mathsf A\) nothing remains. In state \(\mathsf C_j\), \(r\) is valid if and only if \(j\ge2\), with the completing factor \(1\) for \(j=2\) and \(\lambda\) for \(j=3\). In state \(\mathsf O_{00}\), \(r\) is undominated (factor \(\frac12\)); in \(\mathsf O_{01}\) and \(\mathsf O_1\) it is dominated and has at most one sampled neighbor; in \(\mathsf O_2\) and \(\mathsf O_3\) it is a candidate not put into \(\Upsilon\) (factor \(q_2\), respectively \(q_3\)). This gives the displayed sums, whose total is \(\mathrm{fin}(M_r)\).  ∎

For \(p_2=p_3\) the factor \(\lambda\) is \(1\) and the pairs of states \(\mathsf C_2,\mathsf C_3\) and \(\mathsf O_2,\mathsf O_3\) can be merged, which gives an eight-state recursion; in the limit \(p_2=p_3\to0\) the \(\mathsf C\)-coordinates vanish, the states \(\mathsf O_{01}\), \(\mathsf O_1\), \(\mathsf O_2\) and \(\mathsf O_3\) play the role of one state, and the recursion reduces to the three-state recursion of Lemma 7.4.

D.5.3 Light owners and credits

The probabilities in (26)–(28) are ratios of messages at the root, once \(F\) is rooted at the owner or at a neighbor of the exposed vertex. The next lemma compares coordinates of messages; it is the core of the argument. Put \[\mathsf b=2+\frac{2\max(p_2,\,p_3-p_2)}y .\] Throughout this subsection \(0<p_2\le p_3<\frac12\), \(y>0\), \(1\le\varpi_\tau\le2q_\tau\) for \(\tau\in\{2,3\}\), and \(\nu,c_2,c_3:S\to\mathbb Z_{\ge0}\) with \(c_{v,2}+c_{v,3}\le\nu_v\) for all \(v\); these are the standing assumptions on the tree model. Note that \(\varpi_\tau\le2q_\tau\) and \(c_v\le\nu_v\) give \[\tag{29} 2^{-\nu_v}\varpi_2^{c_{v,2}}\varpi_3^{c_{v,3}}\ \le\ 2^{-\nu_v}(2q_2)^{c_{v,2}}(2q_3)^{c_{v,3}}\ \le\ q_2^{c_{v,2}}q_3^{c_{v,3}} ,\] that is, the factor \(\hat h_v\) of Definition D.5.5 is not larger for \(v\notin X\) than for \(v\in X\).

Lemma D.5.9 (Message inequalities).  Let \(F\) be rooted anywhere and let \(f\) be a forcing. Then the messages \(M_v=(A,C_0,\dots,O_3)\) of Definition D.5.7 satisfy:

(a)  \(C_0\ge\frac{p_2}2(O_{00}+O_{01})\) and \(C_j\ge\frac{p_2}2O_j\) for \(j=1,2,3\), for every \(v\) with \(f(v)\ne\mathsf{excl}\);

(b)  \(C_0+C_1+C_2+C_3\le\frac{p_2}y\,A\), for every \(v\) with \(f(v)\ne\mathsf{out}\) none of whose children is excluded;

(c)  \(\chi_{\mathsf S}(M_v)\le\mathsf b\,\chi_{\mathsf O}(M_v)\), for every \(v\) with \(f(v)\ne\mathsf{out}\) none of whose children is excluded;

(d)  \(\chi_{\mathsf S}(M_v)\ge\chi_{\mathsf O}(M_v)\) for every \(v\) with \(f(v)\ne\mathsf{excl}\), and \(\chi_{\mathsf S}(M_v)\ge q_3\,\chi_{\mathsf O}(M_v)\) for every \(v\) with \(f(v)=\mathsf{excl}\).

Lean proof: rsc_messages (formalisation)

Proof. All messages are nonnegative by Lemma D.5.8. We prove (a) and (b) by induction along the merges that produce \(M_v\), and then derive (c) and (d).

(a) For an initial message with \(f(v)=\mathsf{in}\) all coordinates involved are \(0\). Otherwise \(C_1=C_2=C_3=O_1=O_2=O_3=O_{01}=0\), \(C_0=\frac{p_2}2q_2^{c_{v,2}}q_3^{c_{v,3}}\) and \(O_{00}=2^{-\nu_v}\varpi_2^{c_{v,2}}\varpi_3^{c_{v,3}}\), so \(C_0\ge\frac{p_2}2O_{00}\) by (29). For a merge with a child message \(Q\ge0\), the merge formulas give \[\begin{gathered} C_0'=C_0\,\chi^{\mathrm u}_{\mathsf C},\qquad O_{00}'+O_{01}'=(O_{00}+O_{01})(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O}),\\ C_1'=C_1\chi^{\mathrm u}_{\mathsf C}+C_0\chi^{\mathrm s}_{\mathsf C},\qquad O_1'=O_1(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O})+(O_{00}+O_{01})\chi^{\mathrm s}_{\mathsf O},\\ C_2'=C_2\chi^{\mathrm u}_{\mathsf C}+C_1\chi^{\mathrm s}_{\mathsf C},\qquad O_2'=O_2(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O})+O_1\chi^{\mathrm s}_{\mathsf O},\\ C_3'=C_3(\chi^{\mathrm u}_{\mathsf C}+\chi^{\mathrm s}_{\mathsf C})+C_2\chi^{\mathrm s}_{\mathsf C},\qquad O_3'=O_3(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O}+\chi^{\mathrm s}_{\mathsf O})+O_2\chi^{\mathrm s}_{\mathsf O}, \end{gathered}\] and \(\chi^{\mathrm u}_{\mathsf C}-(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O})=c_2+\lambda c_3+\frac12o_{00}\ge0\), \(\chi^{\mathrm s}_{\mathsf C}=2\chi^{\mathrm s}_{\mathsf O}\ge\chi^{\mathrm s}_{\mathsf O}\). Hence each primed \(\mathsf C\)-coordinate is at least \(\frac{p_2}2\) times the corresponding primed \(\mathsf O\)-coordinate, term by term; for instance \(C_1'\ge\frac{p_2}2O_1(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O})+\frac{p_2}2(O_{00}+O_{01})\cdot2\chi^{\mathrm s}_{\mathsf O}\ge\frac{p_2}2O_1'\).

(b) If \(f(v)=\mathsf{in}\), the \(\mathsf C\)-coordinates of the initial message are \(0\), and they stay \(0\) under merging; if \(f(v)=\mathsf{excl}\), both sides are \(0\). Otherwise the initial message has \(C_0=\frac{p_2}2q_2^{c_{v,2}}q_3^{c_{v,3}}=\frac{p_2}y A\). For a merge with a child message \(Q\), summing the second to fifth coordinates of \(P\circ Q\) gives \[C_0'+C_1'+C_2'+C_3'=(C_0+C_1+C_2+C_3)(\chi^{\mathrm u}_{\mathsf C}+\chi^{\mathrm s}_{\mathsf C}),\qquad A'=A\,\chi_{\mathsf S}(Q),\] and \[\chi_{\mathsf S}(Q)-(\chi^{\mathrm u}_{\mathsf C}+\chi^{\mathrm s}_{\mathsf C})=(2c_1-p_2\,o_1)+\bigl(2(\lambda-1)c_2-(p_3-p_2)\,o_2\bigr)\ \ge\ 0\] by (a) for the child, which is not excluded: \(2c_1\ge p_2o_1\), and \(2(\lambda-1)c_2\ge(\lambda-1)p_2o_2=(p_3-p_2)o_2\). So the inequality is preserved.

(c) For \(M_v=(a,c_0,c_1,c_2,c_3,o_{00},o_{01},o_1,o_2,o_3)\), \[\chi_{\mathsf S}(M_v)-2\chi_{\mathsf O}(M_v)=2c_1+2(\lambda-1)c_2-o_{01}-(2-q_2)o_1-(2q_2-q_3)o_2-q_3o_3\ \le\ 2\max(1,\lambda-1)(c_1+c_2),\] since \(2q_2\ge q_3\). By (b), \(c_1+c_2\le\frac{p_2}ya\le\frac{p_2}y\chi_{\mathsf O}(M_v)\), since all terms of \(\chi_{\mathsf O}\) are nonnegative, and \(2\max(1,\lambda-1)p_2=2\max(p_2,p_3-p_2)\). Hence \(\chi_{\mathsf S}(M_v)\le\mathsf b\,\chi_{\mathsf O}(M_v)\).

(d) In the same notation, \[\chi_{\mathsf S}(M_v)-\chi_{\mathsf O}(M_v)=a+\tfrac12o_{00}+\lambda c_3+(2c_1-p_2o_1)+\bigl((2\lambda-1)c_2-(p_3-p_2)o_2\bigr),\] which is nonnegative by (a) if \(f(v)\ne\mathsf{excl}\), because \((2\lambda-1)c_2\ge(2\lambda-1)\frac{p_2}2o_2=(p_3-\frac{p_2}2)o_2\ge(p_3-p_2)o_2\). If \(f(v)=\mathsf{excl}\), then \(a=c_0=\dots=c_3=0\) (the initial message has these coordinates \(0\), and they stay \(0\)), so \[\chi_{\mathsf S}(M_v)-q_3\chi_{\mathsf O}(M_v)=\bigl(1-\tfrac{q_3}2\bigr)o_{00}+(1-q_3)o_{01}+(q_2-q_3)o_1+q_3(1-q_2)o_2+q_3(1-q_3)o_3\ \ge\ 0 .\]  ∎

Definition D.5.10 (Light vertices).  A pair \((\nu,d)\) of nonnegative integers is \(3\)-light if \[2^{-\nu}\mathsf b^{-d}\ \ge\ \Bigl(1+\frac{p_3}y\Bigr)\frac{\varepsilon_3\,y}{\delta_3} ,\] and \(2\)-light if \[2^{-\nu}\mathsf b^{-d}\ \ge\ \frac{\varepsilon_2\,y}{\delta_2} ;\] otherwise it is \(3\)-failing, respectively \(2\)-failing. Given an owner map, a vertex \(v\in S\) is \(\tau\)-light if \((\nu_v,d_F(v))\) is \(\tau\)-light, and \(\tau\)-failing otherwise.

Since \(\varepsilon_3\ge\varepsilon_2\) and \(\delta_3\le\delta_2\), a \(3\)-light pair is \(2\)-light. The right-hand sides are positive because \(q_3>\frac12\); both are small when \(p_3\) is small, because \(\varepsilon_\tau\approx p_\tau\).

The following lemma is the place where the shape of the tree enters. Let \(T_u\) be the smallest subtree of \(F\) that contains \(N_S(u)\). If \(u\) is shared, \(T_u\) has at least two vertices, and its leaves lie in \(N_S(u)\); otherwise a leaf outside \(N_S(u)\) could be removed.

Lemma D.5.11 (Nonnegative correction at light owners).  Under the standing assumptions of Appendix D.5.3 (in particular \(y>0\)), in the setting of Definition D.5.5, let \(u\) be a shared vertex of type \(\tau\) whose owner \(o=o(u)\) is \(\tau\)-light and is a leaf of \(T_u\). Then \(\mathbb{E}_{\mathrm{tm}}[\ln\tilde h_u]\ge0\).

Lean proof: rsc_jlemma (formalisation)

Proof. The proof decomposes the event \(\mathbf 1_u=1\) into disjoint events, each given by a forcing that uses only the values \(\mathsf{in}\) and \(\mathsf{out}\) and forces neither \(o\) nor any \(F\)-neighbor of \(o\) to be \(\mathsf{out}\). For each event it bounds the two probabilities in (26), respectively (27), by the same product of child messages at \(o\), using Lemma D.5.9; the lightness of \(o\) then gives the sign.

The other neighbors lie in one branch. Since \(o\) is a leaf of \(T_u\), it has exactly one neighbor \(w\) in \(T_u\), and \(T_u-o\) is a tree that contains \(w\) and all of \(N'_u=N_S(u)\setminus\{o\}\). So \(N'_u\) lies in the component of \(F-o\) that contains \(w\), and \(w\) is the only \(F\)-neighbor of \(o\) in that component. Hence \(N'_u\cap N_F(o)\subseteq\{w\}\).

Decomposition of the event \(\mathbf 1_u=1\). Let \(N^{\circ}_u=N'_u\setminus\{w\}=\{v_1,\dots,v_h\}\) in any order, so no \(v_i\) is an \(F\)-neighbor of \(o\). For \(1\le i\le h\) let \(f_i\) be the forcing with \(f_i(v_i)=\mathsf{in}\) and \(f_i(v_j)=\mathsf{out}\) for \(j<i\); if \(w\in N'_u\), let \(f_{h+1}\) be the forcing with \(f_{h+1}(w)=\mathsf{in}\) and \(f_{h+1}(v_j)=\mathsf{out}\) for all \(j\le h\). The events “\((X',\Upsilon)\) respects \(f_i\)” are pairwise disjoint, and their union is the event \(N'_u\cap X'\ne\emptyset\), that is, \(\mathbf 1_u=1\): a sampled vertex of \(N^{\circ}_u\) is matched by the first such \(v_i\), and otherwise \(w\) is the sampled vertex of \(N'_u\). No forcing \(f_i\) forces an \(F\)-neighbor of \(o\) to be \(\mathsf{out}\), none forces \(o\), and none excludes a vertex. If \(\tau=2\), then \(N'_u\) consists of one vertex and there is a single forcing \(f_1\).

One forced event. Fix one of the forcings \(f=f_i\), and root \(F\) at \(o\); its children are the \(d=d_F(o)\) neighbors of \(o\). Let \(M_o=(A,C_0,\dots,O_3)\) be the message of \(o\) under \(f\). By Lemma D.5.8, \[\begin{gathered} W^{\mathrm{tm}}\Pr_{\mathrm{tm}}\bigl(f,\ o\in X'\bigr)=A,\qquad W^{\mathrm{tm}}\Pr_{\mathrm{tm}}\bigl(f,\ o\in X\bigr)=A+C_2+\lambda C_3,\\ W^{\mathrm{tm}}\Pr_{\mathrm{tm}}\bigl(f,\ o\notin X\bigr)=\tfrac12O_{00}+O_{01}+O_1+q_2O_2+q_3O_3 , \end{gathered}\] where “\(f\)” denotes the event that \((X',\Upsilon)\) respects \(f\). We bound all three by the product \(\mathrm{pr}_o=\prod_{w'}\chi_{\mathsf O}(M_{w'})\) over the children \(w'\) of \(o\). Since \(o\) is not forced, the merge formulas give \[\begin{gathered} A=q_2^{c_{o,2}}q_3^{c_{o,3}}\,\frac y2\prod_{w'}\chi_{\mathsf S}(M_{w'})\ \le\ \frac y2\prod_{w'}\chi_{\mathsf S}(M_{w'}),\\ O_{00}+O_{01}+O_1+O_2+O_3=2^{-\nu_o}\varpi_2^{c_{o,2}}\varpi_3^{c_{o,3}}\,\mathrm{pr}_o\ \ge\ 2^{-\nu_o}\,\mathrm{pr}_o ; \end{gathered}\] the first inequality holds because \(q_\tau\le1\), the second because \(\varpi_\tau\ge1\), and the identity holds because the sum of the last five coordinates of \(P\circ Q\) is \((O_{00}+O_{01}+O_1+O_2+O_3)\,\chi_{\mathsf O}(Q)\). No child \(w'\) of \(o\) is forced \(\mathsf{out}\) and no vertex is excluded, so Lemma D.5.9(c) gives \(\chi_{\mathsf S}(M_{w'})\le\mathsf b\,\chi_{\mathsf O}(M_{w'})\), and Lemma D.5.9(b) at \(o\) gives \(C_2+\lambda C_3\le\lambda(C_0+C_1+C_2+C_3)\le\frac{\lambda p_2}yA=\frac{p_3}yA\). Since \(\frac12,1,q_2,q_3\ge\frac12\), \[A\le\frac y2\,\mathsf b^{d}\,\mathrm{pr}_o,\qquad A+C_2+\lambda C_3\le\Bigl(1+\frac{p_3}y\Bigr)\frac y2\,\mathsf b^{d}\,\mathrm{pr}_o,\qquad \tfrac12O_{00}+O_{01}+O_1+q_2O_2+q_3O_3\ge\frac12\,2^{-\nu_o}\,\mathrm{pr}_o .\] If \(\tau=3\), then, as \((\nu_o,d)\) is \(3\)-light, \[\delta_3\bigl(\tfrac12O_{00}+O_{01}+O_1+q_2O_2+q_3O_3\bigr)-\varepsilon_3(A+C_2+\lambda C_3) \ \ge\ \frac{\mathrm{pr}_o\,\mathsf b^d}2\Bigl(\delta_3\,2^{-\nu_o}\mathsf b^{-d}-\varepsilon_3\Bigl(1+\frac{p_3}y\Bigr)y\Bigr)\ \ge\ 0 ;\] if \(\tau=2\), then, as \((\nu_o,d)\) is \(2\)-light, \[\delta_2\bigl(\tfrac12O_{00}+O_{01}+O_1+q_2O_2+q_3O_3\bigr)-\varepsilon_2A \ \ge\ \frac{\mathrm{pr}_o\,\mathsf b^d}2\Bigl(\delta_2\,2^{-\nu_o}\mathsf b^{-d}-\varepsilon_2\,y\Bigr)\ \ge\ 0 .\]

Conclusion. Dividing by \(W^{\mathrm{tm}}\) and summing over the disjoint events \(f_i\) gives \(\delta_3\,\Pr_{\mathrm{tm}}(\mathbf 1_u=1,\ o\notin X)\ge\varepsilon_3\,\Pr_{\mathrm{tm}}(\mathbf 1_u=1,\ o\in X)\) if \(\tau=3\), and \(\delta_2\,\Pr_{\mathrm{tm}}(\mathbf 1_u=1,\ o\notin X)\ge\varepsilon_2\,\Pr_{\mathrm{tm}}(\mathbf 1_u=1,\ o\in X')\) if \(\tau=2\); then (26), respectively (27), gives \(\mathbb{E}_{\mathrm{tm}}[\ln\tilde h_u]\ge0\).  ∎

The credits rest on a lower bound for the odds that a vertex of \(S\) is sampled, given that some other vertices are outside the closed sample.

Lemma D.5.12 (Odds of being sampled).  In the setting of Definition D.5.5, let \(U\subseteq S\) and \(w\in S\setminus U\) with \(|U\cap N_F(w)|\le1\), and let \(E\) be the event that \(U\cap X=\emptyset\). Then \[\Pr_{\mathrm{tm}}\bigl(E,\ w\in X'\bigr)\ \ge\ \frac y2\,q_3^{|U\cap N_F(w)|}\,\Pr_{\mathrm{tm}}\bigl(E,\ w\notin X\bigr).\]

Lean proof: rsc_odds, odds1 (formalisation)

Proof. The event \(E\) is the event that \((X',\Upsilon)\) respects the forcing \(f\) with \(f(v)=\mathsf{excl}\) for \(v\in U\). Root \(F\) at \(w\) and let \(M_w=(A,C_0,\dots,O_3)\) be its message under \(f\); \(w\) itself is not forced. By Lemma D.5.8 and the identities used in the proof of Lemma D.5.11, \[\begin{gathered} W^{\mathrm{tm}}\Pr_{\mathrm{tm}}(E,\ w\in X')=A=\frac y2\,q_2^{c_{w,2}}q_3^{c_{w,3}}\prod_{x}\chi_{\mathsf S}(M_x),\\ W^{\mathrm{tm}}\Pr_{\mathrm{tm}}(E,\ w\notin X)\le O_{00}+\dots+O_3=2^{-\nu_w}\varpi_2^{c_{w,2}}\varpi_3^{c_{w,3}}\prod_{x}\chi_{\mathsf O}(M_x), \end{gathered}\] the products being over the children \(x\) of \(w\), that is, over \(N_F(w)\). By Lemma D.5.9(d), \(\chi_{\mathsf S}(M_x)\ge\chi_{\mathsf O}(M_x)\) for \(x\notin U\) and \(\chi_{\mathsf S}(M_x)\ge q_3\chi_{\mathsf O}(M_x)\) for \(x\in U\), and by (29) \(q_2^{c_{w,2}}q_3^{c_{w,3}}\ge2^{-\nu_w}\varpi_2^{c_{w,2}}\varpi_3^{c_{w,3}}\). The claim follows.  ∎

Lemma D.5.13 (Credits).  Put \(\vartheta=\frac y2q_3\) and \(\rho_1=(1+\vartheta)^{-1}\). Under the standing assumptions of Appendix D.5.3, in the setting of Definition D.5.5, let \(u\) be an exposed vertex of type \(\tau\) with owner \(o\). Then \[\Pr_{\mathrm{tm}}\bigl(N'_u\cap X\ne\emptyset\bigm|o\notin X\bigr)\ \ge\ \begin{cases}1-\rho_1&\text{if }\tau=2,\\ 1-\rho_1^2&\text{if }\tau=3. \end{cases}\] Consequently, if \[\tag{30} \ln\varpi_2\le(1-\rho_1)\,\delta_2\qquad\text{and}\qquad \ln\varpi_3\le(1-\rho_1^2)\,\delta_3 ,\] then \(\mathbb{E}_{\mathrm{tm}}[\ln\tilde h^{\mathrm{ex}}_u]\ge0\) for every exposed vertex \(u\).

Lean proof: rsc_credit, rsc_credit_ex (formalisation)

Proof. Let \(E_0\) be the event \(o\notin X\). For \(w\in N'_u\), Lemma D.5.12 with \(U=\{o\}\) gives \(\Pr(E_0,w\in X')\ge\vartheta\Pr(E_0,w\notin X)\), where we write \(\Pr\) for \(\Pr_{\mathrm{tm}}\). Since \(\Pr(E_0,w\in X')\le\Pr(E_0)-\Pr(E_0,w\notin X)\), this gives \((1+\vartheta)\Pr(E_0,w\notin X)\le\Pr(E_0)\), that is, \[\tag{31} \Pr\bigl(w\notin X\bigm|E_0\bigr)\le\rho_1 .\] If \(\tau=2\), then \(N'_u=\{w\}\) and the claim is (31). If \(\tau=3\), choose \(w_1,w_2\in N'_u\) with \(\mathrm{dist}_F(o,w_1)\le\mathrm{dist}_F(o,w_2)\), and let \(E_1\) be the event \(o,w_1\notin X\), which has positive probability. We apply Lemma D.5.12 to \(w=w_2\) and \(U=\{o,w_1\}\): both \(o\) and \(w_1\) cannot be \(F\)-neighbors of \(w_2\), for then the path \(o,w_2,w_1\) would give \(\mathrm{dist}_F(o,w_1)=2>1=\mathrm{dist}_F(o,w_2)\). So \(|U\cap N_F(w_2)|\le1\), and as above \(\Pr(w_2\notin X\mid E_1)\le\rho_1\). Hence \[\Pr\bigl(w_1\notin X,\ w_2\notin X\bigm|E_0\bigr)=\Pr\bigl(w_1\notin X\bigm|E_0\bigr)\Pr\bigl(w_2\notin X\bigm|E_1\bigr)\le\rho_1^2 ,\] and \(\Pr(N'_u\cap X\ne\emptyset\mid E_0)\ge1-\rho_1^2\). The last assertion follows from (28): its right-hand side is at least \(\Pr(E_0)\bigl(\delta_\tau(1-\rho_1^{\tau-1})-\ln\varpi_\tau\bigr)\ge0\) under (30).  ∎

The certificate needs one restriction at the leaves of \(F\), which follows from the minimality of \(S\).

Lemma D.5.14 (Private neighbors of leaves).  If \(S\) is a minimum connected dominating set with \(|S|\ge2\), then every leaf \(v\) of \(F\) has a private neighbor, that is, some \(u\in O\) with \(N_S(u)=\{v\}\).

Lean proof: rsc_leaf (formalisation)

Proof. Since \(F-v\) is a spanning tree of \(S\setminus\{v\}\ne\emptyset\), the set \(S\setminus\{v\}\) induces a connected subgraph, and it dominates \(v\), which is adjacent to its \(F\)-neighbor. If every outside neighbor of \(v\) had a second neighbor in \(S\), then \(S\setminus\{v\}\) would also dominate \(O\), so it would be a connected dominating set of size \(|S|-1\), contradicting minimality.  ∎

The certificate of Appendix D.5.4 bounds \(W^{\mathrm{tm}}\) for the following class of quadruples.

Definition D.5.15 (Admissible quadruples).  A quadruple \((F,\nu,c_2,c_3)\) of a tree \(F\) and maps \(\nu,c_2,c_3:V(F)\to\mathbb Z_{\ge0}\) is admissible if, for every \(v\in V(F)\) with \(d=d_F(v)\) and \(c_v=c_{v,2}+c_{v,3}\): \(c_v\le\nu_v\); \(c_{v,\tau}=0\) whenever \((\nu_v,d)\) is \(\tau\)-light or \(d=0\), for \(\tau\in\{2,3\}\); and \(c_v\le\nu_v-1\) whenever \(d=1\) and \(c_v\ge1\).

The analysis assigns owners and classifies them as follows. A leaf of \(T_u\) always exists in \(N_S(u)\), so the rule is well defined.

Definition D.5.16 (Owner rule).  Choose an owner map \(o\) such that \(o(u)\) is a leaf of \(T_u\) for every shared \(u\) (for a private \(u\) the owner is its unique neighbor in \(S\)). Let \(\nu_v\) be the resulting numbers of owned vertices, let \(O^{\mathrm{co}}\) be the set of shared vertices \(u\) whose owner is \(\tau(u)\)-light and \(O^{\mathrm{ex}}\) the set of shared vertices \(u\) whose owner is \(\tau(u)\)-failing. So \(c_{v,\tau}=0\) if \(v\) is \(\tau\)-light, and \(c_{v,\tau}\) is the number of shared vertices of type \(\tau\) owned by \(v\) if \(v\) is \(\tau\)-failing.

Corollary D.5.17 (Reduction to the tree model).  Assume the standing assumptions of Appendix D.5.3 (in particular \(y>0\)). Let \(\varpi_2,\varpi_3\) satisfy \(1\le\varpi_\tau\le2q_\tau\) and (30), let \(S\) be a minimum connected dominating set with \(|S|=s\), and let \(o\), \(O^{\mathrm{co}}\), \(O^{\mathrm{ex}}\), \(\nu\), \(c_2\), \(c_3\) be given by Definition D.5.16. Then \((F,\nu,c_2,c_3)\) is admissible, \(s+\sum_{v\in S}\nu_v=n\), and \(W^{\mathrm{cl}}_S(y)\ge W^{\mathrm{tm}}(F,\nu,c)\).

Lean proof: rsc_reduction, rsc_reduction_of, exists_ownerOK (formalisation)

Proof. For \(u\in O^{\mathrm{co}}\), the owner \(o(u)\) is \(\tau(u)\)-light and is a leaf of \(T_u\), so Lemma D.5.11 gives \(\mathbb{E}_{\mathrm{tm}}[\ln\tilde h_u]\ge0\); for \(u\in O^{\mathrm{ex}}\), Lemma D.5.13 gives \(\mathbb{E}_{\mathrm{tm}}[\ln\tilde h^{\mathrm{ex}}_u]\ge0\). So Lemma D.5.6 gives the inequality. The identity \(\sum_{v\in S}\nu_v=n-s\) is from Definition D.5.3. For admissibility, \(c_v\le\nu_v\) because the exposed vertices owned by \(v\) are among the vertices it owns, and \(c_{v,\tau}=0\) for \(\tau\)-light \(v\) by Definition D.5.16. Now let \(c_v\ge1\). Then \(v\) owns a shared vertex \(u\), so \(|N_S(u)|\ge2\), hence \(s\ge2\) and \(d_F(v)\ge1\), as \(F\) is a tree with at least two vertices. If \(d_F(v)=1\), then \(v\) is a leaf of \(F\), and Lemma D.5.14 gives a private neighbor \(u'\) of \(v\); by Definition D.5.16 \(v\) owns \(u'\), and \(u'\) is not shared, so at most \(\nu_v-1\) of the vertices owned by \(v\) are exposed.  ∎

D.5.4 The antinorm certificate

The last step bounds \(W^{\mathrm{tm}}(F,\nu,c)\) from below for every admissible quadruple (Definition D.5.15), by the method of Lemma 7.5: a convex region that the merge maps into itself up to a known scale. The certificate works with eight states rather than ten: it forgets whether a vertex has more than two sampled children, except where a sampled parent makes the count matter. This is a relaxation of the tree model, which never overestimates it.

The relaxed model.

Root \(F\) at \(r\); the relaxed model depends on this choice, and everything below holds for every root. For \(X'\subseteq S\) and a candidate \(v\in C_F(X')\) put \(\tilde p(v)=p_3\) if \(v\ne r\), the parent of \(v\) is in \(X'\) and \(v\) has at least two sampled children, and \(\tilde p(v)=p_2\) otherwise; so \(\tilde p(v)\le p_{k_F(v)}\), since \(\tilde p(v)=p_3\) only if \(k_F(v)\ge3\). Let \(\tilde\pi_{X'}(\Upsilon)=\prod_{v\in\Upsilon}\tilde p(v)\prod_{v\in C_F(X')\setminus\Upsilon}(1-\tilde p(v))\), let \(\widetilde{\mathrm{wt}}(X',\Upsilon)\) be \(\mathrm{wt}(X',\Upsilon)\) of Definition D.5.5 with \(\tilde\pi_{X'}\) in place of \(\pi_{X'}\), and let \(\widetilde W(F,\nu,c)=\sum_{X'}\sum_{\Upsilon\subseteq C_F(X')}\widetilde{\mathrm{wt}}(X',\Upsilon)\). For a message \(P\in\mathbb{R}^8\) indexed by \(\mathsf A,\mathsf C_0,\mathsf C_1,\mathsf C_2,\mathsf O_{00},\mathsf O_{01},\mathsf O_1,\mathsf O_2\), where now \(\mathsf C_2\) and \(\mathsf O_2\) mean “at least two sampled children,” define the eight-state merge \(P\mathbin{\tilde\circ}Q\) by the formula for \(P\circ Q\) of Appendix D.5.2 with the pairs of coordinates \((C_2,C_3)\) and \((O_2,O_3)\) of a parent merged into \(C_2\) and \(O_2\), that is, by the merge formula of Appendix D.5.2 with the functionals \[\begin{gathered} \tilde\chi_{\mathsf S}(Q)=2a+2c_1+2\lambda c_2+o_{00}+o_{01}+q_2o_1+q_3o_2,\\ \tilde\chi^{\mathrm s}_{\mathsf C}(Q)=2a,\qquad \tilde\chi^{\mathrm u}_{\mathsf C}(Q)=2c_2+o_{00}+o_{01}+o_1+q_2o_2,\\ \tilde\chi^{\mathrm s}_{\mathsf O}(Q)=a,\qquad \tilde\chi^{\mathrm x}_{\mathsf O}(Q)=c_2,\qquad \tilde\chi^{\mathrm u}_{\mathsf O}(Q)=\tfrac12o_{00}+o_{01}+o_1+q_2o_2, \end{gathered}\] the fourth coordinate \(C_2(\tilde\chi^{\mathrm u}_{\mathsf C}+\tilde\chi^{\mathrm s}_{\mathsf C})+C_1\tilde\chi^{\mathrm s}_{\mathsf C}\), the eighth coordinate \(O_2(\tilde\chi^{\mathrm u}_{\mathsf O}+\tilde\chi^{\mathrm x}_{\mathsf O}+\tilde\chi^{\mathrm s}_{\mathsf O})+O_1\tilde\chi^{\mathrm s}_{\mathsf O}\), and \[\begin{gathered} \widetilde{\mathrm{fin}}(P)=A+C_2+\tfrac12O_{00}+O_{01}+O_1+q_2O_2,\\ \widetilde{\mathrm{Init}}(\nu;c_2,c_3)=\bigl(q_2^{c_2}q_3^{c_3}\tfrac y2,\ q_2^{c_2}q_3^{c_3}\tfrac{p_2}2,\ 0,\ 0,\ 2^{-\nu}\varpi_2^{c_2}\varpi_3^{c_3},\ 0,\ 0,\ 0\bigr). \end{gathered}\] The relaxed messages are \(\widetilde M_v=\widetilde{\mathrm{Init}}(\nu_v;c_{v,2},c_{v,3})\mathbin{\tilde\circ}\widetilde M_{w_1}\mathbin{\tilde\circ}\cdots\mathbin{\tilde\circ}\widetilde M_{w_d}\), for any orders of the children, merged from left to right as in Definition D.5.7; no forcing is used. In the merge, the closure probability \(p_3\) enters only through the terms \(2\lambda c_2\), with \(\lambda=p_3/p_2\), and \(q_3o_2\) of \(\tilde\chi_{\mathsf S}\): a child with at least two sampled children and a sampled parent is closed with probability \(p_3\), and everywhere else the probability is \(p_2\).

Lemma D.5.18 (The relaxed recursion).  Under the standing assumptions of Appendix D.5.3, for every tree \(F\) and every choice of the root \(r\), \(\widetilde W(F,\nu,c)\le W^{\mathrm{tm}}(F,\nu,c)\) and \(\widetilde W(F,\nu,c)=\widetilde{\mathrm{fin}}(\widetilde M_r)\).

Lean proof: rsc_relaxed (formalisation)

Proof. For the inequality, fix \(X'\subseteq S\) and put \(g(\Upsilon)=y^{|X'|}2^{-\Psi_F(X'\cup\Upsilon)}\prod_{v\in S}\hat h_v(X'\cup\Upsilon)\) for \(\Upsilon\subseteq C_F(X')\), so that \(\sum_\Upsilon\pi_{X'}(\Upsilon)g(\Upsilon)\) and \(\sum_\Upsilon\tilde\pi_{X'}(\Upsilon)g(\Upsilon)\) are the contributions of \(X'\) to \(W^{\mathrm{tm}}\) and to \(\widetilde W\). The function \(g\) is nondecreasing under inclusion: let \(v\in C_F(X')\setminus\Upsilon\) and \(X=X'\cup\Upsilon\). Adding \(v\) to \(X\) does not increase the number of vertices of \(S\) with \(N_F[\cdot]\cap X=\emptyset\), and it decreases \(\operatorname{comp}(F[X])\), because the at least two \(F\)-neighbors of \(v\) in \(X'\subseteq X\) lie in distinct components of \(F[X]\) (a path between two of them in \(F[X]\) together with \(v\) would be a cycle of \(F\)); so \(2^{-\Psi_F}\) does not decrease. The factor \(\hat h_v\) changes from \(2^{-\nu_v}\varpi_2^{c_{v,2}}\varpi_3^{c_{v,3}}\) to \(q_2^{c_{v,2}}q_3^{c_{v,3}}\), which is not smaller by (29), and no other factor changes. Since \(\pi_{X'}\) and \(\tilde\pi_{X'}\) are product distributions on the subsets of \(C_F(X')\) with \(\tilde p(v)\le p_{k_F(v)}\) for every \(v\), the monotone coupling of Appendix D.5.1 gives \(\sum_\Upsilon\tilde\pi_{X'}(\Upsilon)g(\Upsilon)\le\sum_\Upsilon\pi_{X'}(\Upsilon)g(\Upsilon)\); summing over \(X'\) gives \(\widetilde W\le W^{\mathrm{tm}}\).

For the identity, the proof of Lemma D.5.8 applies with \(\tilde p(v)\) in place of \(p_{k_F(v)}\) and with the empty forcing: the factor of a vertex \(w\) in \(\Upsilon\) or in \(C_F(X')\setminus\Upsilon\) is now \(\tilde p(w)\) or \(1-\tilde p(w)\), which is \(p_3\), respectively \(q_3\), exactly when \(w\) has a sampled parent and at least two sampled children, and \(p_2\), respectively \(q_2\), otherwise; so a vertex has to count its sampled children only up to two, the states \(\mathsf C_2,\mathsf C_3\) and \(\mathsf O_2,\mathsf O_3\) can be merged, and in the case analysis of the merge the completing factors become those of \(\tilde\chi_{\mathsf S}\), \(\tilde\chi^{\mathrm u}_{\mathsf C}\) and \(\tilde\chi^{\mathrm u}_{\mathsf O}\), \(\tilde\chi^{\mathrm x}_{\mathsf O}\) above (a child in \(\mathsf C_2\) or \(\mathsf O_2\) of a sampled parent gets \(\lambda\), respectively \(q_3\); of an unsampled parent it gets \(1\), respectively \(q_2\)). The root has no parent, so \(\tilde p(r)=p_2\), which gives \(\widetilde{\mathrm{fin}}\).  ∎

In the rest of this subsection all messages are eight-dimensional, and we write \(\circ\), \(\mathrm{fin}\) and \(\mathrm{Init}\) for \(\tilde\circ\), \(\widetilde{\mathrm{fin}}\) and \(\widetilde{\mathrm{Init}}\). The region of the certificate must be closed upwards in an order of the states that the merge respects. The order is the reflexive and transitive closure of \[\mathsf C_2\succ\mathsf C_1\succ\mathsf C_0,\qquad \mathsf C_2\succ\mathsf O_{01}\succ\mathsf O_{00}, \qquad \mathsf O_{01}\succ\mathsf O_1\succ\mathsf O_2 ,\] with \(\mathsf A\) incomparable to the other states. Informally, a higher state is never more restrictive: for instance, a vertex in \(\mathsf C_1\) is valid in every situation in which one in \(\mathsf C_0\) is, and a vertex in \(\mathsf O_{01}\) is dominated and has no sampled child, so it imposes nothing on its parent that a vertex in \(\mathsf O_{00}\), \(\mathsf O_1\) or \(\mathsf O_2\) does not. A set \(U\) of states is an up-set if \(\mathsf x\in U\) and \(\mathsf x'\succ\mathsf x\) imply \(\mathsf x'\in U\). Let \[\mathcal K_{\mathrm{pos}}=\Bigl\{P\in\mathbb{R}^8:\ \sum_{\mathsf x\in U}P_{\mathsf x}\ge0\ \text{for every up-set }U\Bigr\},\] and write \(P\succeq P'\) if \(P-P'\in\mathcal K_{\mathrm{pos}}\). Let \(e_{\mathsf x}\) denote the unit vector of the state \(\mathsf x\). The next lemma collects the properties of \(\mathcal K_{\mathrm{pos}}\) that the certificate uses: a merge with a nonnegative message, on either side, maps \(\mathcal K_{\mathrm{pos}}\) into itself, and \(\mathrm{fin}\) is nonnegative on \(\mathcal K_{\mathrm{pos}}\). As for the cumulative cone of Lemma D.4.1, both properties are checked on a finite set of generators.

Lemma D.5.19 (The poset cone).  Let \(0<p_2\le p_3<1\). The cone \(\mathcal K_{\mathrm{pos}}\) defined above, together with the eight-state merge \(\circ\) and the functional \(\mathrm{fin}\), has the following properties.

(a)  \(\mathcal K_{\mathrm{pos}}\) is a closed convex cone containing \(\mathbb{R}^8_{\ge0}\), and it is the convex cone generated by the eight unit vectors \(e_{\mathsf x}\) and the six vectors \(e_{\mathsf x'}-e_{\mathsf x}\) for the covering pairs \(\mathsf x'\succ\mathsf x\) displayed above.

(b)  If \(Q\in\mathbb{R}^8_{\ge0}\) and \(\Delta\in\mathcal K_{\mathrm{pos}}\), then \(\Delta\circ Q\in\mathcal K_{\mathrm{pos}}\) and \(Q\circ\Delta\in\mathcal K_{\mathrm{pos}}\).

(c)  \(\mathrm{fin}(\Delta)\ge0\) for every \(\Delta\in\mathcal K_{\mathrm{pos}}\).

Lean proof: rsc_cone_a, rsc_cone_b, rsc_cone_c (formalisation)

Proof. (a) \(\mathcal K_{\mathrm{pos}}\) is an intersection of closed half-spaces through \(0\), hence a closed convex cone, and it contains the orthant. Each of the fourteen vectors lies in \(\mathcal K_{\mathrm{pos}}\): the up-set sums of \(e_{\mathsf x}\) are \(0\) or \(1\), and so are those of \(e_{\mathsf x'}-e_{\mathsf x}\) with \(\mathsf x'\succ\mathsf x\), because an up-set containing \(\mathsf x\) contains \(\mathsf x'\). So the cone they generate is contained in \(\mathcal K_{\mathrm{pos}}\).

For the converse, note that the seven states other than \(\mathsf A\) form a rooted tree with root \(\mathsf C_2\), in which \(\mathsf C_1\) and \(\mathsf O_{01}\) are the children of \(\mathsf C_2\), \(\mathsf C_0\) is the child of \(\mathsf C_1\), \(\mathsf O_{00}\) and \(\mathsf O_1\) are the children of \(\mathsf O_{01}\), and \(\mathsf O_2\) is the child of \(\mathsf O_1\); and \(\mathsf x\prec\mathsf x'\) holds exactly when \(\mathsf x'\) is a proper ancestor of \(\mathsf x\). The up-sets are the sets closed under taking parents, with \(\mathsf A\) added or not. Let \(P\in\mathcal K_{\mathrm{pos}}\). The up-set \(\{\mathsf A\}\) gives \(P_{\mathsf A}\ge0\), and \(P-P_{\mathsf A}e_{\mathsf A}\) again lies in \(\mathcal K_{\mathrm{pos}}\), so we may assume \(P_{\mathsf A}=0\). We show by induction on \(|T|\) that a vector \(P\) supported on a subtree \(T\) containing the root, whose sums over the subsets of \(T\) closed under taking parents are nonnegative, is a nonnegative combination of the vectors \(e_{\mathsf x}\) with \(\mathsf x\in T\) and \(e_{\mathsf y}-e_{\mathsf x}\) with \(\mathsf x\in T\) and parent \(\mathsf y\). If \(T=\{\mathsf C_2\}\), then \(P=P_{\mathsf C_2}e_{\mathsf C_2}\) with \(P_{\mathsf C_2}\ge0\). Otherwise let \(\mathsf x\) be a leaf of \(T\) with parent \(\mathsf y\), and put \(P'=P-P_{\mathsf x}e_{\mathsf x}\) if \(P_{\mathsf x}\ge0\) and \(P'=P-(-P_{\mathsf x})(e_{\mathsf y}-e_{\mathsf x})\) otherwise. In both cases \(P'\) is supported on \(T-\mathsf x\), and \(P-P'\) is a nonnegative multiple of a generator. Let \(U'\subseteq T-\mathsf x\) be closed under taking parents; it is also closed under taking parents in \(T\), as \(\mathsf x\) is a leaf. The sum of \(P'\) over \(U'\) equals the sum of \(P\) over \(U'\), except in the second case with \(\mathsf y\in U'\), where it equals the sum of \(P\) over \(U'\cup\{\mathsf x\}\), which is again closed under taking parents. So all these sums are nonnegative, and the induction hypothesis applies to \(P'\) and \(T-\mathsf x\).

(b) The operation \(\circ\) is bilinear and \(\mathcal K_{\mathrm{pos}}\) is a convex cone, so by (a) it suffices to check the fourteen generators \(\Delta\). For a unit vector \(\Delta=e_{\mathsf x}\) both products are nonnegative, since \(\circ\) maps nonnegative messages to nonnegative messages. For the six generators \(e_{\mathsf x'}-e_{\mathsf x}\) we read both products off the merge formula, with \(Q=(a,c_0,c_1,c_2,o_{00},o_{01},o_1,o_2)\) and with the functionals \(\tilde\chi_{\mathsf S},\tilde\chi^{\mathrm s}_{\mathsf C},\dots\) evaluated at \(Q\), where they are nonnegative (we drop the tildes): \[\begin{aligned} (e_{\mathsf C_2}-e_{\mathsf C_1})\circ Q&=\chi^{\mathrm u}_{\mathsf C}\,(e_{\mathsf C_2}-e_{\mathsf C_1}),\\ (e_{\mathsf C_1}-e_{\mathsf C_0})\circ Q&=\chi^{\mathrm u}_{\mathsf C}\,(e_{\mathsf C_1}-e_{\mathsf C_0})+\chi^{\mathrm s}_{\mathsf C}\,(e_{\mathsf C_2}-e_{\mathsf C_1}),\\ (e_{\mathsf C_2}-e_{\mathsf O_{01}})\circ Q&=(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O})(e_{\mathsf C_2}-e_{\mathsf O_{01}}) +\chi^{\mathrm s}_{\mathsf O}\bigl[(e_{\mathsf C_2}-e_{\mathsf O_{01}})+(e_{\mathsf O_{01}}-e_{\mathsf O_1})\bigr]\\ &\qquad+\bigl(a+c_2+\tfrac12o_{00}\bigr)e_{\mathsf C_2},\\ (e_{\mathsf O_{01}}-e_{\mathsf O_{00}})\circ Q&=\chi^{\mathrm u}_{\mathsf O}\,(e_{\mathsf O_{01}}-e_{\mathsf O_{00}}),\\ (e_{\mathsf O_{01}}-e_{\mathsf O_1})\circ Q&=(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O})(e_{\mathsf O_{01}}-e_{\mathsf O_1})+\chi^{\mathrm s}_{\mathsf O}\,(e_{\mathsf O_1}-e_{\mathsf O_2}),\\ (e_{\mathsf O_1}-e_{\mathsf O_2})\circ Q&=(\chi^{\mathrm u}_{\mathsf O}+\chi^{\mathrm x}_{\mathsf O})(e_{\mathsf O_1}-e_{\mathsf O_2}), \end{aligned}\] and \[\begin{aligned} Q\circ(e_{\mathsf C_2}-e_{\mathsf C_1})&=(2(\lambda-1)a,\,2c_0,\,2c_1,\,2c_2,\,0,\,o_{00}+o_{01},\,o_1,\,o_2),\\ Q\circ(e_{\mathsf C_1}-e_{\mathsf C_0})&=2a\,e_{\mathsf A},\\ Q\circ(e_{\mathsf C_2}-e_{\mathsf O_{01}})&=((2\lambda-1)a,\,c_0,\,c_1,\,c_2,\,0,\,0,\,0,\,0)+o_{00}\,(e_{\mathsf O_{01}}-e_{\mathsf O_{00}}),\\ Q\circ(e_{\mathsf O_{01}}-e_{\mathsf O_{00}})&=\tfrac12\,(0,\,0,\,0,\,0,\,o_{00},\,o_{01},\,o_1,\,o_2),\\ Q\circ(e_{\mathsf O_{01}}-e_{\mathsf O_1})&=p_2\,a\,e_{\mathsf A},\\ Q\circ(e_{\mathsf O_1}-e_{\mathsf O_2})&=\bigl((p_3-p_2)a,\,p_2c_0,\,p_2c_1,\,p_2c_2,\,p_2o_{00},\,p_2o_{01},\,p_2o_1,\,p_2o_2\bigr). \end{aligned}\] In the third line of the first list, the coefficient of \(e_{\mathsf C_2}\) in \((e_{\mathsf C_2}-e_{\mathsf O_{01}})\circ Q\) is \(\chi^{\mathrm u}_{\mathsf C}+\chi^{\mathrm s}_{\mathsf C}\), and \(\chi^{\mathrm u}_{\mathsf C}+\chi^{\mathrm s}_{\mathsf C}-\chi_{\mathsf O}=a+c_2+\frac12o_{00}\). In the second list, the entries involving \(\lambda\) and \(p_3\) come from the terms \(2\lambda c_2\) and \(q_3o_2\) of \(\tilde\chi_{\mathsf S}\); they are nonnegative because \(\lambda\ge1\) and \(p_3\ge p_2\). So every product is a nonnegative combination of generators, and it lies in \(\mathcal K_{\mathrm{pos}}\).

(c) \(\mathrm{fin}\) is linear with the coefficients \((1,0,0,1,\frac12,1,1,q_2)\), so by (a) it suffices that it is nonnegative on the generators: its values on the unit vectors are these coefficients, and its values on the six vectors \(e_{\mathsf x'}-e_{\mathsf x}\) are \(1\), \(0\), \(0\), \(\frac12\), \(0\) and \(p_2\), in the order of the covering pairs displayed above.  ∎

For a finite nonempty set \(Z\subseteq\mathbb{R}^8_{\ge0}\) let \(\mathcal P_Z=\mathrm{conv}(Z)+\mathcal K_{\mathrm{pos}}\) and \(c_Z=\min_{z\in Z}\mathrm{fin}(z)\). For \(\mathcal Y\subseteq\mathbb{R}^8\) and \(P\in\mathbb{R}^8\) we write \(P\in\mathcal Y+\mathcal K_{\mathrm{pos}}\) also as “\(P\) dominates a point of \(\mathcal Y\).” Write \(\mathrm{Init}(\nu)=\mathrm{Init}(\nu;0,0)\) and \(\mathrm{Init}(\infty)=(\frac y2,\frac{p_2}2,0,0,0,0,0,0)\), the limit of \(\mathrm{Init}(\nu)\) as \(\nu\to\infty\), and let \[L(\nu)=\bigl\{(c_2,c_3)\in\mathbb Z_{\ge0}^2:\ 1\le c_2+c_3\le\nu\bigr\}\] be the exposure lattice. The initial message is linear in the pair \((q_2^{c_2}q_3^{c_3},\ \varpi_2^{c_2}\varpi_3^{c_3})\): \[\tag{32} \mathrm{Init}(\nu;c_2,c_3)=q_2^{c_2}q_3^{c_3}\,\mathrm{Init}(\infty)+\varpi_2^{c_2}\varpi_3^{c_3}\,2^{-\nu}e_{\mathsf O_{00}} .\]

The next lemma states the conditions that the certificate verifies. The first two conditions are the analogues of the two conditions of the antinorm certificate for RS-\(\theta\) (Lemma 7.5), with one scale \(\hat e\) for solution vertices and owned vertices in place of \(e_1,e_2\), so that the bound \(c_Z\hat e^n\) does not depend on the solution fraction \(s/n\) and no case distinction on \(s/n\) is needed; the remaining four control the initial messages of vertices with exposures, which the proof places in a scaled copy of \(\mathcal P_Z\) either directly or after their first few merges.

Lemma D.5.20 (Certificate conditions).  Let \(0<p_2\le p_3<\frac12\), \(y>0\), \(1\le\varpi_\tau\le2q_\tau\) and \(0<\hat e<q_3\). Let \(Z\subseteq\mathbb{R}^8_{\ge0}\) be finite and nonempty, let \(\nu_0\ge0\) and \(\nu_E\ge1\) be integers, and let \(d^{(3)}_{\min},d^{(2)}_{\min}:\{1,\dots,\nu_E\}\to\mathbb Z_{\ge1}\) with \(d^{(3)}_{\min}\le d^{(2)}_{\min}\); put \(j_\tau(\nu)=\max(1,d^{(\tau)}_{\min}(\nu)-1)\). Suppose that:

(C1)  \(z\circ z'\in\mathcal P_Z\) for all \(z,z'\in Z\);

(C2)  \(\mathrm{Init}(\nu)\in \hat e^{1+\nu}\mathcal P_Z\) for \(0\le\nu\le\nu_0\), and \(\mathrm{Init}(\infty)\in \hat e^{\nu_0+2}\mathcal P_Z\);

(C3)  for \(1\le\nu\le\nu_E\) and \(\tau\in\{2,3\}\), every pair \((\nu,d)\) with \(1\le d<d^{(\tau)}_{\min}(\nu)\) is \(\tau\)-light;

(C4)  for \(2\le\nu\le\nu_E\): if \(d^{(3)}_{\min}(\nu)\le1\), then \(\mathrm{Init}(\nu;0,c_3)\in \hat e^{1+\nu}\mathcal P_Z\) for \(1\le c_3\le\nu-1\), and if moreover \(d^{(2)}_{\min}(\nu)\le1\), then \(\mathrm{Init}(\nu;c_2,c_3)\in \hat e^{1+\nu}\mathcal P_Z\) for all \((c_2,c_3)\in L(\nu-1)\) with \(c_2\ge1\);

(C5)  for \(1\le\nu\le\nu_E\) and \((c_2,c_3)\in L(\nu)\), with \(j=j_3(\nu)\) if \(c_2=0\) and \(j=j_2(\nu)\) if \(c_2\ge1\): \(\hat e^{-1-\nu}\mathrm{Init}(\nu;c_2,c_3)\circ Q_1\circ\cdots\circ Q_j\in\mathcal P_Z\) (merged from left to right) for all \(Q_1,\dots,Q_j\in\mathcal P_Z\cap\mathbb{R}^8_{\ge0}\);

(C6)  \((q_3/\hat e)^{\nu_E+1}\ge \hat e^{-(\nu_0+1)}\).

Then every admissible quadruple \((F,\nu,c_2,c_3)\) in which \(F\) has \(s\) vertices satisfies, with \(\nu(F)=\sum_{v\in V(F)}\nu_v\), \[W^{\mathrm{tm}}(F,\nu,c)\ \ge\ \widetilde W(F,\nu,c)\ \ge\ c_Z\,\hat e^{\,s+\nu(F)} .\]

Lean proof: certificate_conditions, certificate_conditions_of, C5_of_stages, rsc_tree_bound, rsc_relaxed (formalisation)

Proof. The first inequality is Lemma D.5.18, for any root. Step 1 shows that membership in scaled copies of \(\mathcal P_Z\) is supermultiplicative under the merge. Step 2 places the initial message \(\mathrm{Init}(\nu_v;c_{v,2},c_{v,3})\) of most vertices \(v\) in \(\hat e^{1+\nu_v}\mathcal P_Z\), and Step 3 treats the remaining vertices, whose message reaches the right scaled copy only after its first few merges. Step 4 propagates this up the tree and reads off the bound at the root.

Step 1: supermultiplicativity. The merge \(\circ\), the cone \(\mathcal K_{\mathrm{pos}}\) and the set \(Z\) satisfy the hypotheses of Lemma D.4.2: \(\circ\) is bilinear; by Lemma D.5.19(a), \(\mathcal K_{\mathrm{pos}}\) is a convex cone containing \(\mathbb{R}^8_{\ge0}\); by Lemma D.5.19(b), \(\Delta\circ Q\) and \(Q\circ\Delta\) lie in \(\mathcal K_{\mathrm{pos}}\) for \(\Delta\in\mathcal K_{\mathrm{pos}}\) and \(Q\in\mathbb{R}^8_{\ge0}\); and \(z\circ z'\in\mathcal P_Z\) for \(z,z'\in Z\) by (C1). Hence Lemma D.4.2 gives \(\mathcal P_Z+\mathcal K_{\mathrm{pos}}=\mathcal P_Z\), \(\eta\mathcal P_Z\subseteq\eta'\mathcal P_Z\) for \(0<\eta'\le\eta\), and (its part (b) with \(Y=Y'=Z\)) \(P\circ Q\in\eta\eta'\mathcal P_Z\) whenever \(P\in\eta\mathcal P_Z\), \(Q\in\eta'\mathcal P_Z\) and \(Q\ge0\). In particular, if \(P\ge0\) and \(P\in\eta\mathcal P_Z\), then \(\eta''P\in\eta''\eta\,\mathcal P_Z\subseteq\eta\mathcal P_Z\) for every \(\eta''\ge1\).

Step 2: initial messages. Let \((F,\nu,c_2,c_3)\) be admissible and \(v\in V(F)\), with \(\nu=\nu_v\), \((c_2,c_3)=(c_{v,2},c_{v,3})\), \(c=c_2+c_3\) and \(d=d_F(v)\). We claim that \(\mathrm{Init}(\nu;c_2,c_3)\in \hat e^{1+\nu}\mathcal P_Z\), unless \(c\ge1\), \(\nu\le\nu_E\) and \(d\ge2\); that remaining case is handled in Step 3.

• Case \(c=0\) and \(\nu\le\nu_0\): this is (C2).

• Case \(c=0\) and \(\nu>\nu_0\): \(\mathrm{Init}(\nu)-\mathrm{Init}(\infty)=2^{-\nu}e_{\mathsf O_{00}}\in\mathcal K_{\mathrm{pos}}\), so \(\mathrm{Init}(\nu)\in \hat e^{\nu_0+2}\mathcal P_Z+\mathcal K_{\mathrm{pos}}=\hat e^{\nu_0+2}\mathcal P_Z\subseteq \hat e^{1+\nu}\mathcal P_Z\) by Step 1, as \(\hat e^{1+\nu}\le \hat e^{\nu_0+2}\) (since \(\hat e<1\)).

• Case \(c\ge1\) and \(\nu>\nu_E\): by (32), \(\mathrm{Init}(\nu;c_2,c_3)-q_3^\nu\mathrm{Init}(\infty)=(q_2^{c_2}q_3^{c_3}-q_3^\nu)\mathrm{Init}(\infty)+\varpi_2^{c_2}\varpi_3^{c_3}2^{-\nu}e_{\mathsf O_{00}}\ge0\), because \(q_2^{c_2}q_3^{c_3}\ge q_3^{c}\ge q_3^\nu\); so \(\mathrm{Init}(\nu;c_2,c_3)\succeq q_3^\nu\mathrm{Init}(\infty)\). Moreover \(\hat e^{-1-\nu}q_3^\nu=\hat e^{-1}(q_3/\hat e)^\nu\ge \hat e^{-1}(q_3/\hat e)^{\nu_E+1}\ge \hat e^{-\nu_0-2}\) by (C6), as \(q_3/\hat e>1\). So \(\hat e^{-1-\nu}\mathrm{Init}(\nu;c_2,c_3)\) dominates a multiple \(\eta''\,\hat e^{-\nu_0-2}\mathrm{Init}(\infty)\) with \(\eta''\ge1\), which lies in \(\mathcal P_Z\) by (C2) and Step 1.

• Case \(c\ge1\), \(\nu\le\nu_E\) and \(d\le1\): admissibility excludes \(d=0\) and gives \(c\le\nu-1\) for \(d=1\), so \(\nu\ge2\); it also says that \((\nu,1)\) is \(\tau\)-failing for every \(\tau\) with \(c_\tau\ge1\), so by (C3) \(d^{(\tau)}_{\min}(\nu)\le1\) for these \(\tau\). If \(c_2=0\), then \(c_3\ge1\), \(d^{(3)}_{\min}(\nu)\le1\) and the first part of (C4) applies. If \(c_2\ge1\), then \(d^{(2)}_{\min}(\nu)\le1\), hence also \(d^{(3)}_{\min}(\nu)\le1\), and \((c_2,c_3)\in L(\nu-1)\), so the second part of (C4) applies.

Step 3: failing internal vertices with few owned vertices. Let \(c\ge1\), \(\nu\le\nu_E\) and \(d\ge2\). If \(c_2\ge1\), then \((\nu,d)\) is \(2\)-failing by admissibility, so \(d\ge d^{(2)}_{\min}(\nu)\) by (C3) and \(d-1\ge\max(1,d^{(2)}_{\min}(\nu)-1)=j_2(\nu)\); if \(c_2=0\), then \((\nu,d)\) is \(3\)-failing, and likewise \(d-1\ge j_3(\nu)\). Let \(j\) be \(j_2(\nu)\) or \(j_3(\nu)\) accordingly. With \(F\) rooted as in Step 4, \(v\) has at least \(d-1\ge j\) children \(w_1,\dots,w_j,\dots\), in the order of merging. Suppose that \(\widetilde M_{w_i}\in\eta_i\mathcal P_Z\) with \(\eta_i>0\) for \(i\le j\) (this will be the induction hypothesis of Step 4). The messages \(\widetilde M_{w_i}/\eta_i\) are nonnegative and lie in \(\mathcal P_Z\), so (C5) gives, after \(j\) merges and by bilinearity, \[\mathrm{Init}(\nu;c_2,c_3)\circ\widetilde M_{w_1}\circ\cdots\circ\widetilde M_{w_j}\ \in\ \hat e^{1+\nu}\eta_1\cdots\eta_j\,\mathcal P_Z .\]

Step 4: induction over the tree. Root \(F\) at any vertex \(r\), and let \(\eta(F_v)=\hat e^{|V(F_v)|+\nu(F_v)}\) with \(\nu(F_v)=\sum_{x\in V(F_v)}\nu_x\). We show \(\widetilde M_v\in\eta(F_v)\mathcal P_Z\) by induction from the leaves. If Step 2 applies to \(v\), start from \(\mathrm{Init}(\nu_v;c_{v,2},c_{v,3})\in \hat e^{1+\nu_v}\mathcal P_Z\) and merge the children one by one; each merge with a child \(w\) multiplies the scale by \(\eta(F_w)\), by Step 1, as \(\widetilde M_w\ge0\). Otherwise Step 3 handles the first \(j\) children and Step 1 the remaining ones. At the root, \(\widetilde M_r=\hat e^{s+\nu(F)}(z+\Delta)\) with \(z\in\mathrm{conv}(Z)\) and \(\Delta\in\mathcal K_{\mathrm{pos}}\), so by Lemma D.5.18 and Lemma D.5.19(c) \[\widetilde W(F,\nu,c)=\mathrm{fin}(\widetilde M_r)=\hat e^{s+\nu(F)}\bigl(\mathrm{fin}(z)+\mathrm{fin}(\Delta)\bigr)\ \ge\ \hat e^{s+\nu(F)}c_Z ,\] since \(\mathrm{fin}\) is linear and \(\mathrm{fin}(z)\ge c_Z\) for \(z\in\mathrm{conv}(Z)\).  ∎

Condition (C5) is established by the certificate in two steps. First, for a point \(P\ge0\) and finite sets \(Y_1,\dots,Y_{j-1}\subseteq\mathbb{R}^8_{\ge0}\) (the stages), if every product \(x\circ z\) with \(x\in Y_{i-1}\) and \(z\in Z\) lies in \(\mathrm{conv}(Y_i)+\mathcal K_{\mathrm{pos}}\) for \(i=1,\dots,j\), where \(Y_0=\{P\}\) and \(Y_j=Z\), then \(P\circ Q_1\circ\cdots\circ Q_j\in\mathcal P_Z\) for all \(Q_i\in\mathcal P_Z\cap\mathbb{R}^8_{\ge0}\): by induction on \(i\), Lemma D.4.2(b) applied to the sets \((Y_{i-1},Z,Y_i)\) in place of \((Y,Z,Y')\), with \(\eta=\eta'=1\), gives \(P\circ Q_1\circ\cdots\circ Q_i\in\mathrm{conv}(Y_i)+\mathcal K_{\mathrm{pos}}\) (the closure hypothesis of that lemma was checked in Step 1 above). Second, the property in (C5) is preserved under convex combinations of \(P\), because \(P\mapsto P\circ Q_1\circ\cdots\circ Q_j\) is linear and \(\mathcal P_Z\) is convex; it is preserved when \(j\) is increased, by Step 1; and by (32), \(\mathrm{Init}(\nu;c_2,c_3)\) for \((c_2,c_3)\in L(\nu)\) is a convex combination of the points \(\mathrm{Init}(\nu;c_2',c_3')\) over the vertices \((c'_2,c'_3)\) of the convex hull of \(\bigl\{(q_2^{c_2}q_3^{c_3},\varpi_2^{c_2}\varpi_3^{c_3}):0\le c_2+c_3\le\nu\bigr\}\), with the same coefficients that express the pair of \((c_2,c_3)\) through the hull vertices; the vertex \((0,0)\), if used, contributes \(\mathrm{Init}(\nu)\), which lies in \(\hat e^{1+\nu}\mathcal P_Z\) by the first two cases of Step 2. Since \(j_3(\nu)\le j_2(\nu)\), it therefore suffices to provide stages for \(\hat e^{-1-\nu}\mathrm{Init}(\nu;0,c_3)\) with \(j=j_3(\nu)\) for \(1\le c_3\le\nu\) and for \(\hat e^{-1-\nu}\mathrm{Init}(\nu;c_2,c_3)\) with \(j=j_2(\nu)\) for the hull vertices with \(c_2\ge1\). The same convexity argument reduces the second part of (C4) to the hull vertices of the lattice for \(\nu-1\).

The certificate verifies the conditions of Lemma D.5.20 and the hypotheses of Corollary D.5.17 for the parameters of RSC. Condition (C3) does not involve the certificate: it follows from Lemma D.5.21, which can be checked by hand and is also checked in Lean (Appendix F.9).

Lemma D.5.21 (Light pairs for RSC).  For \(p_2=0.07\), \(p_3=0.077\) and \(y=31/100\) the pairs \((1,1)\), \((1,2)\), \((2,1)\) and \((3,1)\) are \(3\)-light, and the pairs \((1,3)\) and \((2,2)\) are \(2\)-light.

Lean proof: rsc_lightpairs, rsc_C3 (formalisation)

Proof. Here \(q_2=\frac{93}{100}\), \(q_3=\frac{923}{1000}\), \(\max(p_2,p_3-p_2)=p_2\) and \(\mathsf b=2+\frac{14}{31}=\frac{76}{31}\). Since \(-\ln(1-p)=\sum_{k\ge1}p^k/k\le p+\frac{p^2}2+\frac{p^3}{3(1-p)}\) for \(0<p<1\), we have \(\varepsilon_3\le\frac{443756467}{5538000000}<0.08013\) and \(\varepsilon_2\le\frac{404957}{5580000}<0.07258\). Since \(\ln\frac{1+z}{1-z}=2\sum_{k\ge0}\frac{z^{2k+1}}{2k+1}\ge2z+\frac23z^3\) for \(0\le z<1\), with \(z=\frac{423}{1423}\) we have \(\delta_3=\ln\frac{1846}{1000}>0.61202\), and with \(z=\frac13\) and four terms, \(\delta_2=\ln2>0.69313\). Hence, with \(1+\frac{p_3}y=\frac{387}{310}\), \[\Bigl(1+\frac{p_3}y\Bigr)\frac{\varepsilon_3\,y}{\delta_3}<\frac{387}{310}\cdot\frac{0.08013\cdot0.31}{0.61202}<0.0507,\qquad \frac{\varepsilon_2\,y}{\delta_2}<\frac{0.07258\cdot0.31}{0.69313}<0.0325 .\] The left-hand side \(2^{-\nu}\mathsf b^{-d}=2^{-\nu}(\frac{31}{76})^d\) is decreasing in \(\nu\) and \(d\), and its values are \(\frac{961}{11552}>0.0831\) at \((1,2)\), \(\frac{31}{304}>0.1019\) at \((2,1)\), \(\frac{31}{608}>0.0509\) at \((3,1)\), which gives the \(3\)-light pairs, and \(\frac{29791}{877952}>0.0339\) at \((1,3)\) and \(\frac{961}{23104}>0.0415\) at \((2,2)\), which gives the \(2\)-light pairs.  ∎

Lemma D.5.22 (Credits for RSC).  For \(p_2=0.07\), \(p_3=0.077\), \(y=31/100\), \(\varpi_2=\frac{272657}{250000}=1.090628\) and \(\varpi_3=\frac{288677}{250000}=1.154708\), we have \(1\le\varpi_\tau\le2q_\tau\) and (30) holds.

Lean proof: rsc_rsccredits (formalisation) · Certificates: rsc-certificate.json.gz, rsc-verify.py, rsc_verify_core.py, rsc_modelg.py, rsc-credits.py, rsc-verify-independent.py

Proof. \(2q_2=1.86\) and \(2q_3=1.846\). Here \(\vartheta=\frac{31}{200}\cdot\frac{923}{1000}\) and \(\rho_1=\frac{200000}{228613}\), so \(1-\rho_1=\frac{28613}{228613}\) and \(1-\rho_1^2=\frac{12263903769}{52263903769}\). The two inequalities of (30) read \(\ln\varpi_2\le\frac{28613}{228613}\ln2\) and \(\ln\varpi_3\le\frac{12263903769}{52263903769}(\ln2+\ln q_3)\); numerically the two sides are \(0.086753678\) versus \(0.086753685\) and \(0.143847499\) versus \(0.143847506\) (left-hand sides rounded up, right-hand sides rounded down), so the credits were chosen as large as the inequalities allow, to six decimals. They follow from the truncated-series bounds \(\ln\varpi_\tau\le\sum_{k=1}^{15}(-1)^{k+1}(\varpi_\tau-1)^k/k\), an alternating series with decreasing terms cut after a positive term; \(\ln2\ge2\sum_{k=0}^{6}3^{-2k-1}/(2k+1)\); and \(-\ln q_3\le\sum_{k=1}^{6}p_3^k/k+p_3^7/(7q_3)\): with these bounds, evaluated exactly in rational arithmetic (Appendix F.4), the right-hand sides exceed the left-hand sides by more than \(6\cdot10^{-9}\) and \(5\cdot10^{-9}\).  ∎

What the certificate proves.

The certificate is a finite set \(Z\) of \(1024\) points of \(\mathbb{R}^8\), and \(\mathcal P_Z=\mathrm{conv}(Z)+\mathcal K_{\mathrm{pos}}\) is the region that they span together with the poset cone. Each of (C1), (C2), (C4) and (C5) is a finite list of statements of the form “this rational vector lies in this region”. Together they say two things: the message of a single vertex lies in a scaled copy of \(\mathcal P_Z\) ((C2), (C4) and (C5), the last after a few merges), and the merge of two messages in scaled copies of \(\mathcal P_Z\) lies in the product scaled copy ((C1), extended from the points of \(Z\) to the whole region by bilinearity and Step 1 of the proof of Lemma D.5.20). By induction from the leaves, the message of every tree, of any size and shape, then lies in \(\hat e^{s+\nu(F)}\mathcal P_Z\), and since \(\mathrm{fin}\ge c_Z\) on \(\mathcal P_Z\), the generating function is at least \(c_Z\hat e^{s+\nu(F)}\). Finitely many exact checks thus prove a bound for arbitrarily large trees, in the same way as an invariant region of a map bounds all of its iterates.

The next proposition records the parameters of the certificate for the remaining conditions and the sizes of its point sets.

Proposition D.5.23 (The certificate).  Let \(p_2=0.07\), \(p_3=0.077\), \(y=31/100\), \(\varpi_2=272657/250000\), \(\varpi_3=288677/250000\), \(\hat e=145729958/189014575\), \(\nu_0=24\), \(\nu_E=36\), \(d^{(3)}_{\min}(1)=3\), \(d^{(3)}_{\min}(2)=d^{(3)}_{\min}(3)=2\), \(d^{(2)}_{\min}(1)=4\), \(d^{(2)}_{\min}(2)=3\), \(d^{(2)}_{\min}(3)=2\) and \(d^{(3)}_{\min}(\nu)=d^{(2)}_{\min}(\nu)=1\) for \(4\le\nu\le36\). There is a set \(Z\) of \(1024\) nonnegative rational points with \(c_Z>0.6060\), and there are nonnegative rational stages for (C5), namely for \(\nu=1\) a set \(Y_1\) of \(62\) points for \((c_2,c_3)=(0,1)\) and sets \(Y_1\), \(Y_2\) of \(62\) and \(304\) points for \((1,0)\), and for \(\nu=2\) sets \(Y_1\) of \(61\), \(62\) and \(62\) points for \((1,0)\), \((1,1)\) and \((2,0)\), such that conditions (C1)–(C6) of Lemma D.5.20 hold.

Lean proof: rsc_cert_full, certZ_cZ_eq, certZ_cZ_digits, rsc_j_one, IsStages, isStages_C5, rsc_cert, certZ_cZ, certZ_C1, certZ_C2, certZ_C4, certZ_C5 (formalisation) · Certificates: rsc-certificate.json.gz, rsc-verify.py, rsc_verify_core.py, rsc_modelg.py, rsc-credits.py, rsc-verify-independent.py

For all other pairs \((\nu,(c_2,c_3))\) we have \(j=1\) in (C5), and no intermediate stages are needed.

Proof. Condition (C3) is Lemma D.5.21, together with the fact, noted after Definition D.5.10, that a \(3\)-light pair is \(2\)-light, and the hypotheses on the credits are Lemma D.5.22. Condition (C6) is an inequality between explicit rationals, \((q_3/\hat e)^{37}\ge \hat e^{-25}\); numerically the two sides are about \(779\) and \(666\). The values of \(d^{(3)}_{\min}\) and \(d^{(2)}_{\min}\) satisfy \(d^{(3)}_{\min}\le d^{(2)}_{\min}\). The certificate file of Appendix F.4 stores the values stated above for \(1\le\nu\le4\) and the value \(0\) in place of \(1\) for \(5\le\nu\le36\), and its check uses these stored values; they give the same conditions: for \(d^{(\tau)}_{\min}(\nu)\in\{0,1\}\) no integer \(d\) satisfies \(1\le d<d^{(\tau)}_{\min}(\nu)\), so (C3) requires nothing at \(\nu\); the hypotheses \(d^{(\tau)}_{\min}(\nu)\le1\) in (C4) hold; and \(j_\tau(\nu)=\max(1,d^{(\tau)}_{\min}(\nu)-1)=1\) in (C5); these are the only places where the hypotheses of Lemma D.5.20 use the values besides \(d^{(3)}_{\min}\le d^{(2)}_{\min}\).

Conditions (C1) and (C2) are finite lists of relations of the form \(P\in\mathrm{conv}(\mathcal Y)+\mathcal K_{\mathrm{pos}}\), with \(\mathcal Y=Z\). By the convexity argument after Lemma D.5.20, (C4) follows from (C1), (C2), the first part of (C4) and the second part at the hull vertices with \(c_2\ge1\) of the lattice for \(\nu-1\), which are again such relations. By the argument on stages and the convexity argument after Lemma D.5.20, (C5) follows from (C1), (C2) and the relations \(x\circ z\in\mathrm{conv}(Y_i)+\mathcal K_{\mathrm{pos}}\) for \(x\in Y_{i-1}\), \(z\in Z\) and \(1\le i\le j\), for the chains \(Y_0=\{P\},Y_1,\dots,Y_j=Z\) that start at the scaled initial messages \(P\) listed there and use the stages of the proposition when \(j\ge2\). So (C1), (C2), (C4) and (C5) follow from finitely many relations, each asserting that a given rational vector \(P\) (a product \(z\circ z'\), a scaled initial message, or a staged product \(x\circ z\)) lies in \(\mathrm{conv}(\mathcal Y)+\mathcal K_{\mathrm{pos}}\) for a given finite set \(\mathcal Y\subseteq\mathbb{R}^8_{\ge0}\). For each such \(P\) the certificate gives nonnegative rational coefficients \(\alpha_i\) with \(\sum_i\alpha_i\ge1\) and points \(z_i\in\mathcal Y\) such that \(P\succeq\sum_i\alpha_iz_i\); this implies \(P\in\mathrm{conv}(\mathcal Y)+\mathcal K_{\mathrm{pos}}\), because \(\sum_i\alpha_iz_i-\sum_i\frac{\alpha_i}{\sum_j\alpha_j}z_i\) is a nonnegative vector. All these relations, the nonnegativity of the points of \(Z\) and of the stages, and the value \(c_Z=0.60605\ldots\) have been verified in exact rational arithmetic (Appendix F.4).  ∎

Proof of Proposition 8.6. Let \(S\) be a minimum connected dominating set and \(s=|S|\). By Lemma D.5.22 and Corollary D.5.17, the owner rule produces an admissible quadruple \((F,\nu,c_2,c_3)\) with \(s+\sum_v\nu_v=n\) and \(W^{\mathrm{cl}}_S(y)\ge W^{\mathrm{tm}}(F,\nu,c)\). By Proposition D.5.23 the hypotheses of Lemma D.5.20 hold (note \(0<\hat e<q_3\), as \(\hat e<0.771<0.923\)), so \(W^{\mathrm{cl}}_S(y)\ge c_Z\,\hat e^n\ge0.6060\,\hat e^n\), since \(c_Z>0.6060\).  ∎

D.6 The black-box barrier

We work in the black-box model of Definition 9.2, with the notation of Section 9.2, and assume \(n\ge1\). Appendix D.6.1 states the lower bounds for mean exponents, which Corollary 9.3 evaluates for Connected Dominating Set, and Appendix D.6.2 proves them together with Corollary 9.3 and Theorem 9.4.

D.6.1 Lower bounds for mean exponents

One sample size.

The first lower bound concerns the hypothesis (5) of Theorem 4.1 at a single sample size \(t\), that is, the class \(\mathcal{C}_{\mathrm{mean}}(t,\lambda)\). For \(h>0\) put \(\mathsf N(h)=\lfloor\lambda\binom kt/h\rfloor+1\) and \[\Sigma(h)=\frac{\binom nt}{\binom kt}\cdot \begin{cases}2^{h}&\text{if }0<h\le\lambda,\\ \min\bigl(2^{h},\mathsf N(h)\bigr)&\text{if }h>\lambda. \end{cases}\] The idea is that the adversary gives touched \(t\)-sets the running-time exponent \(h\) and plants the running-time exponent \(0\) on the untouched ones: as long as fewer than \(\mathsf N(h)\) subsets of the planted solution are touched, the planted instance lies in \(\mathcal{C}_{\mathrm{mean}}(t,\lambda)\). Since \(\binom kt\) is a positive integer, \(\mathsf N(h)>\binom kt\) holds exactly when \(h\le\lambda\); so in the first case the planted instance lies in the class whatever the algorithm touches, and only the \(\mathrm{EXT}\) charges count. In particular \(\Sigma(\lambda)=\mathsf R_t\) when \(\lambda>0\).

Theorem D.6.1 (Black-box lower bound, one sample size).  Let \(0\le t\le k\le n\) and \(\lambda\ge0\), and let \(\mathcal H\) be a deterministic algorithm in the black-box model (Definition 9.2) that is correct for \(\mathcal{C}_{\mathrm{mean}}(t,\lambda)\). For every \(h>0\), the cost of \(\mathcal H\) on some no-instance is at least \(\frac13\min\{\binom nk,\Sigma(h)\}\). In particular, with \(h^{*}=\max\left(1,\lceil\log((\lambda+1)\binom kt)\rceil\right)\), \[\mathrm{cost}(\mathcal H)\ \ge\ \frac13\min\Bigl\{\binom nk,\ \max\Bigl(\mathsf R_t,\ \binom nt\frac{\lambda}{h^{*}}\Bigr)\Bigr\}.\]

So, up to the factor \(\lambda/h^{*}\) and the cap \(\binom nk\), the cost of a deterministic black-box algorithm is at least \(\binom nt\), whereas the randomized algorithm of the mean-exponent theorem has cost \(\mathsf R_t\) up to a polynomial factor; the ratio is exponential in \(n\) whenever \(\log\binom kt\) exceeds \(\lambda\) by a linear amount (Corollary D.6.2). Theorem D.6.1 follows from Theorem D.6.3 below applied to the subclass in which the mean bound at every level other than \(t\) is \(n\), and the proof below proceeds in this way.

The adversary can use integer running-time exponents in \(\{0,\dots,n\}\), as for the number of Steiner terminals in Section 6: whenever \((\lambda+1)\binom kt\le2^{n}\), the second claim of Theorem D.6.1 holds with \(\mathsf R_t\) replaced by \(\mathsf R_t/2\), and by \(\mathsf R_t\) itself for integer \(\lambda\), against algorithms that are correct for the instances of \(\mathcal{C}_{\mathrm{mean}}(t,\lambda)\) with integer running-time exponents (see the paragraph on integer running-time exponents in Appendix D.6.2). We record the gap in quantitative form.

Corollary D.6.2 (Gap).  Let \(\lambda\ge1\) and \(\varepsilon>0\) with \(\log\binom kt\ge\lambda+\varepsilon n\). Every deterministic black-box algorithm that is correct for \(\mathcal{C}_{\mathrm{mean}}(t,\lambda)\) has cost at least \(\min\{\binom nk,2^{\varepsilon n}\mathsf R_t\}/(3(n+2))\) on some no-instance, while the Markov-cutoff algorithm of Remark 4.2, with the exact threshold \(\lambda+1\), decides the class with one-sided error at cost at most \(6(\lambda+1)\mathsf R_t\).

In applications the mean bound usually holds for every sample size, and the lower bound persists under this stronger hypothesis.

All sample sizes.

For \(0\le j\le k\) and \(h>0\) let \(\mathsf N_j(h)\), \(\Sigma_j(h)\) and \(h^{*}_j\) be \(\mathsf N(h)\), \(\Sigma(h)\) and \(h^{*}\) with \((t,\lambda)\) replaced by \((j,\lambda_j)\), and put \(\mathsf R_j=\binom nj\binom kj^{-1}2^{\lambda_j}\). The planting argument applies at all levels at once, because the condition of \(\mathcal{C}_{\mathrm{mean}}^{\mathrm{all}}\) at level \(j\) involves only the \(j\)-subsets of the solution.

Theorem D.6.3 (Black-box lower bound, all sample sizes).  Let \(\lambda_0,\dots,\lambda_k\ge0\), and let \(\mathcal H\) be a deterministic black-box algorithm that is correct for \(\mathcal{C}_{\mathrm{mean}}^{\mathrm{all}}(\lambda_0,\dots,\lambda_k)\). For all \(h_0,\dots,h_k>0\), the cost of \(\mathcal H\) on some no-instance is at least \(\frac13\min\{\binom nk,\min_{0\le j\le k}\Sigma_j(h_j)\}\). In particular, \[\mathrm{cost}(\mathcal H)\ \ge\ \frac13\min\Bigl\{\binom nk,\ \min_{0\le j\le k}\max\Bigl(\mathsf R_j,\ \binom nj\frac{\lambda_j}{h^{*}_j}\Bigr)\Bigr\},\] and \(\lambda_j/h^{*}_j\ge1/(n+2)\) for every \(j\) with \(\lambda_j\ge1\).

D.6.2 Proofs

The planting step.

All lower bounds use the same adversary. Fix a deterministic algorithm \(\mathcal H\) and run it on the no-instance \(\mathcal{F}=\emptyset\), \(g(X,\ell)=h_{|X|}\) for every \(\ell\), where the reals \(h_0,\dots,h_n\ge0\) are chosen later. Since \(\mathcal H\) is deterministic, its run is a fixed sequence of calls. Call a set \(X\) touched if \(\mathrm{COST}\) or \(\mathrm{EXT}\) was called on \((X,\ell)\) for some \(\ell\). Let \(\mathsf T_j\) be the number of touched \(j\)-sets, \(\gamma_j\) and \(e_j\) the numbers of \(\mathrm{COST}\) and \(\mathrm{EXT}\) calls on \(j\)-sets, and \(z\) the number of \(\mathrm{MEM}\) calls. Then \(\mathsf T_j\le \gamma_j+e_j\), and the cost of the run is \[\tag{33} \mathrm{cost}=\sum_j\gamma_j+\sum_je_j2^{h_j}+z .\] For a \(k\)-set \(S\) let \(q_j(S)\) be the number of touched \(j\)-subsets of \(S\), and call \(S\) clean if \(\mathrm{MEM}(S)\) was not called and no \(\mathrm{EXT}\) call was made on a subset of \(S\).

Lemma D.6.4 (Planting).  Let \(S\) be a clean \(k\)-set and let \(g'\colon2^U\times\mathbb{N}\to\mathbb{R}_{\ge0}\) agree with \(g\) on \((X,\ell)\) for every touched \(X\) and every \(\ell\). Then \(\mathcal H\) answers no on the instance \((\{S\},g')\). Moreover, for a uniformly random \(S\in\binom Uk\), \[\begin{gathered} \Pr[\mathrm{MEM}(S)\text{ called}]\le\frac{z}{\binom nk},\qquad \Pr[\text{some EXT call on a subset of }S]\le\sum_je_j\frac{\binom kj}{\binom nj},\\ \text{and}\quad\mathbb{E}\,q_j(S)=\mathsf T_j\frac{\binom kj}{\binom nj}\quad\text{for every }j. \end{gathered}\]

Proof. We show, by induction along the run, that the instance \((\{S\},g')\) answers every call exactly as the no-instance does. Since \(\mathcal H\) is deterministic, it then makes the same calls on both instances and returns the same answer, and on the no-instance that answer is no by correctness. Consider a call of the run. A \(\mathrm{COST}\) call concerns a touched set, on which \(g'\) and \(g\) agree. A call \(\mathrm{MEM}(X)\) answers yes on \((\{S\},g')\) only for \(X=S\), which was not queried, so it answers no on both instances. A call \(\mathrm{EXT}(X,\ell)\) answers no on \((\{S\},g')\) if \(X\not\subseteq S\), which cleanness ensures, so it too answers no on both instances. For the probabilities, the \(z\) calls of \(\mathrm{MEM}\) query at most \(z\) of the \(\binom nk\) equally likely sets; a fixed \(j\)-set lies in \(S\) with probability \(\binom{n-j}{k-j}/\binom nk=\binom kj/\binom nj\), which gives the second bound by the union bound and the expectation by linearity.  ∎

The three lower bounds differ only in which planted instances lie in the class. The next lemma turns this into one counting inequality.

Lemma D.6.5 (Counting).  Let \(\mathcal H\) be correct for a class \(\mathcal{C}\), and let \(M_0,\dots,M_k\ge1\) be integers such that for every clean \(k\)-set \(S\) with \(q_j(S)<M_j\) for all \(j\in\{0,\dots,k\}\) there is a \(g'\) as in Lemma D.6.4 with \((\{S\},g')\in\mathcal{C}\). Then \[\tag{34} 1\le\frac{z}{\binom nk}+\sum_{j=0}^{k}\Bigl(e_j\frac{\binom kj}{\binom nj} +\mathbf 1\bigl[M_j\le\tbinom kj\bigr]\,\frac{\mathsf T_j\binom kj}{M_j\binom nj}\Bigr).\]

Proof. Let \(S\) be a clean \(k\)-set with \(q_j(S)<M_j\) for all \(j\le k\). The instance \((\{S\},g')\) lies in \(\mathcal{C}\) and its family \(\{S\}\) contains a set of size exactly \(k\), so correctness forces \(\mathcal H\) to answer yes on it, which contradicts Lemma D.6.4. Hence, for uniformly random \(S\in\binom Uk\), the event that \(S\) is clean and \(q_j(S)<M_j\) for all \(j\le k\) has probability zero, and the union bound gives \[1\le\Pr[\mathrm{MEM}(S)\text{ called}]+\Pr[\text{some EXT call on a subset of }S] +\sum_{j=0}^{k}\Pr[q_j(S)\ge M_j].\] Lemma D.6.4 bounds the first two terms. For the third, \(q_j(S)\le\binom kj\), so \(\Pr[q_j(S)\ge M_j]=0\) if \(M_j>\binom kj\), and otherwise Markov’s inequality and Lemma D.6.4 give \(\Pr[q_j(S)\ge M_j]\le\mathbb{E}\,q_j(S)/M_j=\mathsf T_j\binom kj/(M_j\binom nj)\).  ∎

We use the following facts about \(\Sigma\).

Lemma D.6.6 (The function \(\Sigma\)).  Let \(0\le t\le k\le n\), \(\lambda\ge0\), and let \(\mathsf N\), \(\Sigma\), \(\mathsf R_t\) and \(h^{*}\) be as in Section 9.2 and Appendix D.6.1.

(a)  \(\Sigma(h)\le\binom nt2^{h}/\binom kt\) for every \(h>0\), and \(\Sigma(h)\le\binom nt\mathsf N(h)/\binom kt\) for every \(h>\lambda\).

(b)  If \(\lambda>0\) then \(\Sigma(\lambda)=\mathsf R_t\); if \(\lambda=0\) then \(\Sigma(h)=\mathsf R_t\) for every \(h>0\). In both cases some \(h>0\) has \(\Sigma(h)=\mathsf R_t\).

(c)  \(\Sigma(h^{*})\ge\binom nt\lambda/h^{*}\).

(d)  If \(\lambda\ge1\) then \(\lambda/h^{*}\ge1/(n+2)\).

Proof. (a) Both cases of the definition of \(\Sigma\) are at most \(\binom nt2^{h}/\binom kt\), and the second case is at most \(\binom nt\mathsf N(h)/\binom kt\).

(b) If \(\lambda>0\), the first case applies at \(h=\lambda\) and gives \(\Sigma(\lambda)=\binom nt2^{\lambda}/\binom kt=\mathsf R_t\). If \(\lambda=0\), every \(h>0\) falls in the second case with \(\mathsf N(h)=1\), so \(\Sigma(h)=\binom nt\binom kt^{-1}\min(2^{h},1)=\binom nt/\binom kt=\mathsf R_t\).

(c) By definition, \(h^{*}\ge1\) and \(2^{h^{*}}\ge(\lambda+1)\binom kt\). If \(h^{*}\le\lambda\), then \(\Sigma(h^{*})=\binom nt2^{h^{*}}/\binom kt\ge\binom nt(\lambda+1)\ge\binom nt\lambda/h^{*}\). If \(h^{*}>\lambda\), then \(\mathsf N(h^{*})\le\lambda\binom kt/h^{*}+1\le(\lambda+1)\binom kt\le2^{h^{*}}\), so \(\Sigma(h^{*})=\binom nt\mathsf N(h^{*})/\binom kt\), and \(\mathsf N(h^{*})>\lambda\binom kt/h^{*}\) gives the claim.

(d) Since \(\lceil x\rceil\le x+1\) and \(\log\binom kt\le n\), \(h^{*}\le\max(1,\log(\lambda+1)+n+1)=\log(\lambda+1)+n+1\). For \(\lambda\ge1\) we have \(2^{\lambda}\ge1+\lambda\) (equality at \(\lambda=1\), and the left side grows faster), that is \(\log(\lambda+1)\le\lambda\), so \(h^{*}\le\lambda+n+1\le(n+2)\lambda\).  ∎

Proof of Theorem D.6.3. Run the planting step with the given values \(h_0,\dots,h_k\) at levels \(0,\dots,k\) and, say, \(h_j=0\) at levels \(j>k\), which play no role, as a set of size \(j>k\) is not a subset of a \(k\)-set. We apply Lemma D.6.5 with \(M_j=\mathsf N_j(h_j)\). Let \(S\) be a clean \(k\)-set with \(q_j(S)<\mathsf N_j(h_j)\) for all \(j\le k\), and define \(g'\) to be \(0\) on the untouched subsets of \(S\) of every size, with every second argument, and \(g\) elsewhere. The hypothesis of \(\mathcal{C}_{\mathrm{mean}}^{\mathrm{all}}\) at level \(j\) involves only \(g'(X,k)\) for \(X\in\binom Sj\); every touched \(j\)-subset of \(S\) has \(g'=h_j\) and every untouched one has \(g'=0\), so the mean of \(g'(\cdot,k)\) over \(\binom Sj\) is \(q_j(S)h_j/\binom kj\le(\mathsf N_j(h_j)-1)h_j/\binom kj\le\lambda_j\). Hence \((\{S\},g')\in\mathcal{C}_{\mathrm{mean}}^{\mathrm{all}}(\lambda_0,\dots,\lambda_k)\), and (34) holds.

We bound the term of (34) at level \(j\). By Lemma D.6.6(a), \(e_j\binom kj/\binom nj\le e_j2^{h_j}/\Sigma_j(h_j)\). The indicator is \(1\) exactly when \(h_j>\lambda_j\), since \(\mathsf N_j(h_j)>\binom kj\) holds exactly when \(h_j\le\lambda_j\); then \(\Sigma_j(h_j)\le\binom nj\mathsf N_j(h_j)/\binom kj\) by Lemma D.6.6(a), so the Markov term is at most \(\mathsf T_j/\Sigma_j(h_j)\). As \(h_j>0\), \(\mathsf T_j+e_j2^{h_j}\le\gamma_j+e_j+e_j2^{h_j}\le2(\gamma_j+e_j2^{h_j})\). So the level-\(j\) term is at most \(2\,\mathrm{cost}_j/\Sigma_j(h_j)\) with \(\mathrm{cost}_j=\gamma_j+e_j2^{h_j}\), and by (33) the right-hand side of (34) is at most \(\mathrm{cost}/\binom nk+2\,\mathrm{cost}/\min_j\Sigma_j(h_j)\le3\,\mathrm{cost}/\min\{\binom nk,\min_j\Sigma_j(h_j)\}\), where \(\mathrm{cost}\) is the cost of the run on the no-instance. This proves the first claim.

For the second claim, choose at each level \(j\) the better of two values of \(h_j\): a value with \(\Sigma_j(h_j)=\mathsf R_j\), which exists by Lemma D.6.6(b), and \(h_j=h^{*}_j\), for which \(\Sigma_j(h^{*}_j)\ge\binom nj\lambda_j/h^{*}_j\) by Lemma D.6.6(c). The first claim with these values gives the displayed bound, and Lemma D.6.6(d) gives \(\lambda_j/h^{*}_j\ge1/(n+2)\) for \(\lambda_j\ge1\).  ∎

Proof of Theorem D.6.1. Let \(\mathcal H\) be correct for \(\mathcal{C}_{\mathrm{mean}}(t,\lambda)\), and let \(\lambda_t=\lambda\) and \(\lambda_j=n\) for \(j\ne t\). Every instance of \(\mathcal{C}_{\mathrm{mean}}^{\mathrm{all}}(\lambda_0,\dots,\lambda_k)\) satisfies the level-\(t\) condition, so it lies in \(\mathcal{C}_{\mathrm{mean}}(t,\lambda)\), and \(\mathcal H\) is correct for the subclass. Apply Theorem D.6.3 with \(h_t=h\) and \(h_j=n\) for \(j\ne t\). At a level \(j\ne t\) we have \(h_j=n\le\lambda_j\), so \(\Sigma_j(n)=\binom nj2^{n}/\binom kj\ge2^{n}\ge\binom nk\), and likewise \(\mathsf R_j=\binom nj2^{n}/\binom kj\ge\binom nk\). Hence in both bounds of Theorem D.6.3 the levels \(j\ne t\) do not lower the minimum below \(\binom nk\), and the bounds read \(\frac13\min\{\binom nk,\Sigma(h)\}\) and \(\frac13\min\{\binom nk,\max(\mathsf R_t,\binom nt\lambda/h^{*})\}\).  ∎

Integer running-time exponents.

In the proof of Theorem D.6.1 the adversary uses the running-time exponents \(n\) at the levels \(j\ne t\) and \(h\) at level \(t\). The second claim uses \(h=h^{*}\), an integer, which is at most \(n\) exactly when \((\lambda+1)\binom kt\le2^{n}\), and a value \(h\) with \(\Sigma(h)=\mathsf R_t\), which is \(h=\lambda\) when \(\lambda>0\) and need not be an integer. An integer replacement in \(\{1,\dots,n\}\) loses at most a factor \(2\): if \(\lambda\ge n\), take \(h=n\le\lambda\), for which \(\Sigma(n)=\binom nt2^{n}/\binom kt\ge2^{n}\ge\binom nk\), so the capped bound \(\frac13\binom nk\) is attained in full; if \(1\le\lambda<n\), take \(h=\lfloor\lambda\rfloor\le\lambda\), for which \(\Sigma(h)=\binom nt2^{\lfloor\lambda\rfloor}/\binom kt\ge\mathsf R_t/2\); and if \(0\le\lambda<1\), take \(h=1>\lambda\), for which \(\mathsf N(1)\ge1\) and \(\Sigma(1)=\binom nt\binom kt^{-1}\min(2,\mathsf N(1))\ge\binom nt/\binom kt\ge\mathsf R_t/2\), as \(2^{\lambda}<2\). So, whenever \((\lambda+1)\binom kt\le2^{n}\), the second claim of Theorem D.6.1 holds with \(\mathsf R_t\) replaced by \(\mathsf R_t/2\) against algorithms that are correct for the instances of \(\mathcal{C}_{\mathrm{mean}}(t,\lambda)\) with integer running-time exponents in \(\{0,\dots,n\}\); the planted running-time exponents are \(0\), so the planted instances have integer running-time exponents as well. If \(\lambda\) is an integer, the three choices give \(\Sigma(h)\ge\mathsf R_t\) or the full cap, so \(\mathsf R_t\) itself remains.

Proof of Corollary D.6.2. First, \(\binom nt=\mathsf R_t\binom kt2^{-\lambda}\ge2^{\varepsilon n}\mathsf R_t\), and \(\lambda/h^{*}\ge1/(n+2)\) by Lemma D.6.6(d), so \(\max(\mathsf R_t,\binom nt\lambda/h^{*})\ge2^{\varepsilon n}\mathsf R_t/(n+2)\), and Theorem D.6.1 gives the lower bound. For the upper bound, computation is free in the model, so the algorithm of Remark 4.2 can use the exact threshold \(\lambda+1\). Each trial then makes one \(\mathrm{COST}\) call and at most one \(\mathrm{EXT}\) call on a set with \(g\le\lambda+1\), so it costs at most \(1+2^{\lambda+1}\le3\cdot2^{\lambda}\). The number of trials is \(\lceil(\lambda+1)\binom nt\binom kt^{-1}\rceil\le2(\lambda+1)\binom nt\binom kt^{-1}\), since the argument of the ceiling is at least \(1\). The total cost is therefore at most \(6(\lambda+1)\mathsf R_t\), and Remark 4.2 gives the one-sided error.  ∎

Restatement of Corollary 9.3.

For its proof we use the function \(f_\sigma\) of (9) with \(\beta=1\), \(\rho=2\) and \(\sigma=\frac12\), that is, \[f_{1/2}(\tau)=H(\tau)-\tfrac12H(2\tau)+1-2\tau\qquad(0\le\tau\le\tfrac12).\]

Proof. The lower bound comes from the second bound of Theorem D.6.3: we estimate \(\mathsf R_j\) and \(\binom nj\) by entropy and show that, at every level, the larger of the two exponents is at least \(H(\frac14)\). The upper bound is the linear-savings corollary.

For \(1\le j\le k\), \(\lambda_j\) is nonincreasing in \(j\) and \(\lambda_k=1\), so every \(\lambda_j\ge1\) (also \(\lambda_0=n\ge1\)), and Theorem D.6.3 gives cost at least \(\frac13\min\{\binom nk,\min_j\max(\mathsf R_j,\binom nj/(n+2))\}\). We bound its terms. First, \(\binom n{n/2}\ge2^{n}/(n+1)\) by (1), and \(\mathsf R_0=2^{n}\). For \(1\le j\le k\) put \(\tau=j/n\in(0,\frac12]\). Since \(\lambda_j=n-1-2j+4j/n\ge n(1-2\tau)-1\), (1) gives \[\mathsf R_j\ge\frac{2^{nH(\tau)}}{n+1}\cdot2^{-kH(j/k)}\cdot2^{n(1-2\tau)-1}=\frac{2^{nf_{1/2}(\tau)}}{2(n+1)}, \qquad\frac{\binom nj}{n+2}\ge\frac{2^{nH(\tau)}}{(n+1)(n+2)} .\] So \[\max\Bigl(\mathsf R_j,\frac{\binom nj}{n+2}\Bigr)\ge\frac{2^{n\max(f_{1/2}(\tau),H(\tau))}}{2(n+1)(n+2)} .\] We show that \(\max(f_{1/2}(\tau),H(\tau))\ge H(\frac14)\) for every \(\tau\in(0,\frac12]\). For \(\tau\ge\frac14\) this holds because \(H\) increases on \([0,\frac12]\). For \(0<\tau\le\frac14\), Remark 4.4 with \((\beta,\rho,\sigma)=(1,2,\frac12)\) and \(p=\frac14\) shows that \(f_{1/2}\) is convex with minimizer \(\tau^*=\frac13\), so it is nonincreasing on \([0,\frac13]\) and \(f_{1/2}(\tau)\ge f_{1/2}(\frac14)=H(\frac14)-\frac12H(\frac12)+1-\frac12=H(\frac14)\). (In terms of binomial coefficients: the ratio of consecutive terms of \(\binom nj\binom kj^{-1}2^{n-2j}\) is \((n-j)/(4(k-j))\), which is at most \(1\) for \(j\le n/3\).) Since \(H(\frac14)<1\), all terms are at least \(2^{nH(1/4)}/(2(n+1)(n+2))\), and Theorem D.6.3 gives the lower bound. Finally \(H(\frac14)=\frac14\log4+\frac34\log\frac43=2-\frac34\log3\), so \(2^{H(1/4)}=4\cdot3^{-3/4}=1.7547\ldots\).

For the upper bound, the class satisfies (7) with \((\beta,\rho,C)=(1,2,1)\) at every sample size: \(\lambda_0=n\le n+1\), and for \(1\le j\le k\), \(\lambda_j=n-1-2j+4j/n\le n-2j+1\) because \(j\le n/2\). By Corollary 4.3(1), Algorithm MLS-\(\theta\) decides the class with one-sided error in time \(2^{C+nF^{*}(1/2)}N^{O(1)}\), where \(F^{*}(\frac12)=1+\frac12\log\frac12+\frac12\log\frac32=\frac12\log3\) by (6), since \(p=\frac14<\frac12\). In the model, a trial of \((t,\theta)\) in Algorithm MLS-\(\theta\) makes one \(\mathrm{COST}\) call and at most one \(\mathrm{EXT}\) call, on a set with \(g\le\theta\), so it costs at most \(2^{\theta+1}\). The algorithm of Corollary 4.3(1) uses \(\Gamma=1\), confidence \(\kappa=1\) and a rate \(\psi\) with \(\psi n\le C+nF^{*}(\frac12)+1\le n+2\). By (2), \(\Theta\le\psi n+3\) and \(m_{t,\theta}2^{\theta+1}\le(2M_{t,\theta}+1)2^{\theta+1} \le16\kappa(\Theta+1)2^{\psi n}+2^{\Theta+1}\le16(\kappa(\Theta+1)+1)2^{\psi n}\). Summing over the \((k+1)(\Theta+1)\) pairs \((t,\theta)\), the cost is at most \(16(k+1)(\Theta+1)(\kappa(\Theta+1)+1)2^{\psi n}\), and \(2^{\psi n}\le4\cdot\sqrt3^{\,n}\), so the cost is \(\sqrt3^{\,n}n^{O(1)}\).  ∎

Restatement of Theorem 9.4.

Proof. Take \(h_j=n\) for all \(j\), so that \(g\equiv n\). Fix a feasible \(t\) and apply Lemma D.6.5 with \(M_t=\binom kt-J_t+1\) and \(M_j=\binom kj+1\) for \(j\ne t\); these are integers, and \(M_t\ge1\) since \(J_t\le\binom kt/2\). Let \(S\) be a clean \(k\)-set with \(q_j(S)<M_j\) for all \(j\le k\); the conditions at levels \(j\ne t\) hold for every \(S\), and the condition at level \(t\) says that at least \(J_t\) sets of \(\binom St\) are untouched. Let \(g'\) be \(0\) on \(J_t\) of them, with every second argument, and \(g\) elsewhere. Then \(\mu_t(S)\ge J_t\ge\binom nt2^{-\psi n}/\Gamma\) under \(g'\), so \((\{S\},g')\in\mathcal{C}_{\mathrm{exp}}(\psi,\Gamma)\). In (34) the indicator vanishes at the levels \(j\ne t\), and \(M_t>\binom kt/2\) at level \(t\), so, using \(\binom kj/\binom nj\le1\le2^{n}/\binom nk\), \[1\le\frac{z}{\binom nk}+\sum_je_j\frac{\binom kj}{\binom nj}+\frac{\mathsf T_t\binom kt}{M_t\binom nt} \le\frac{z+\sum_je_j2^{n}}{\binom nk}+\frac{2\mathsf T_t}{\binom nt}.\] By (33), \(z+\sum_je_j2^{n}\le\mathrm{cost}\) and \(\mathsf T_t\le \gamma_t+e_t\le\mathrm{cost}\), so the right-hand side is at most \(3\,\mathrm{cost}/\min\{\binom nk,\binom nt\}\).  ∎

D.7 Balanced set-inclusion families

This subsection constructs the balanced families of Definition 9.5 and proves the following theorem of Section 9.3, whose notation we keep.

Restatement of Theorem 9.6.

The construction achieves exact balance. Its building block is a covering of ordered structures on small blocks by subsets that are good on every prefix: when a set is ordered by non-increasing weight, a prefix-good subset captures its proportional share of the weight (Lemma D.7.2), and at least a \(1/p\) fraction of all \({\tilde t}\)-subsets of a \(p\)-set is prefix-good (Lemma D.7.1). Formally, for a finite set \(S\) with a linear order \(u_1,\dots,u_p\) and an integer \(0\le {\tilde t}\le p\), call \(Y\subseteq S\) with \(|Y|=\tilde t\) prefix-good if \(p\,|Y\cap\{u_1,\dots,u_i\}|\ge {\tilde t}\,i\) for \(i=1,\dots,p\).

Lemma D.7.1 (Rotation).  Let \(p\ge1\). For every \(\xi\in\{0,1\}^{p}\) with \(\sum_i\xi_i={\tilde t}\), some cyclic rotation of \(\xi\) has \(i\)-th prefix sum at least \(({\tilde t}/p)i\) for every \(i\). Consequently, for a fixed order of \(S\), a uniformly random \({\tilde t}\)-subset of \(S\) is prefix-good with probability at least \(1/p\).

Proof. Let \(\pi_0=0\) and \(\pi_i=\sum_{l\le i}(\xi_l-{\tilde t}/p)\), so \(\pi_p=0\). Pick \(m\in\{0,\dots,p-1\}\) minimizing \(\pi_m\). The rotation that starts at \(\xi_{m+1}\) has \(i\)-th prefix sum minus \(({\tilde t}/p)i\) equal to \(\pi_{m+i}-\pi_m\ge0\) if \(m+i\le p\), and to \(\pi_p+\pi_{m+i-p}-\pi_m=\pi_{m+i-p}-\pi_m\ge0\) if \(m+i>p\). For the second claim, identify \({\tilde t}\)-subsets of \(S\) with their indicator vectors. The cyclic group of rotations acts on these vectors and the uniform distribution is invariant under it. Each orbit has size dividing \(p\) and contains a prefix-good member, so at least a \(1/p\) fraction of every orbit, and hence of all \({\tilde t}\)-subsets, is prefix-good.  ∎

The second ingredient is the weight guarantee of prefix-good sets.

Lemma D.7.2 (Abel summation).  If \(w\ge0\) is non-increasing along the order \(u_1,\dots,u_p\) of \(S\) and \(Y\subseteq S\) is prefix-good with \(|Y|=\tilde t\), then \(w(Y)\ge({\tilde t}/p)w(S)\).

Proof. Put \(c_i=|Y\cap\{u_1,\dots,u_i\}|\), \(c_0=0\) and \(w(u_{p+1})=0\). Then \(w(Y)=\sum_{i=1}^{p}w(u_i)(c_i-c_{i-1})=\sum_{i=1}^{p}c_i\bigl(w(u_i)-w(u_{i+1})\bigr)\). Each difference is nonnegative and \(c_i\ge({\tilde t}/p)i\), so \(w(Y)\ge({\tilde t}/p)\sum_{i=1}^{p}i\bigl(w(u_i)-w(u_{i+1})\bigr)=({\tilde t}/p)\sum_{i=1}^pw(u_i)=({\tilde t}/p)w(S)\).  ∎

The block families are found by greedy Set Cover, whose guarantee we use in the following form: on an instance whose sets each cover at most \(\Delta\) elements, greedy returns a cover of size at most \(1+\ln \Delta\) times the optimum (Johnson 1974; Lovász 1975).

Lemma D.7.3 (Block families).  Let \(\iota\ge1\), let \(q_1,\dots,q_{\mathsf r}\ge0\) be integers with \(\sum_jq_j\le\iota\), and let \(\tilde t_j\) be integers with \(0\le \tilde t_j\le q_j\) and \({\tilde t}=\sum_j\tilde t_j\). A deterministic algorithm computes, in time polynomial in \(\iota!\,2^{\iota}\), an \((\iota;q;{\tilde t})\)-balanced family \(\mathcal B\) over \([\iota]\) with \[|\mathcal B|\le N_{\mathrm{cov}}\,(1+\ln\iota!)^{2},\qquad N_{\mathrm{cov}}=\binom\iota {\tilde t}\prod_j\frac{\max(q_j,1)}{\binom{q_j}{\tilde t_j}} .\]

Proof. An ordered structure is a tuple of pairwise disjoint sets \(S_j\subseteq[\iota]\) with \(|S_j|=q_j\), each with a linear order; let \(\mathcal O\) be the set of ordered structures, so that \(|\mathcal O|=\iota!/(\iota-\sum_jq_j)!\le\iota!\). A candidate \(X\in\binom{[\iota]}{\tilde t}\) covers a structure if \(|X\cap S_j|=\tilde t_j\) for every \(j\) (so \(X\subseteq\bigcup_jS_j\), as \(\sum_j\tilde t_j={\tilde t}=|X|\)) and \(X\cap S_j\) is prefix-good in \(S_j\) for every \(j\) with \(q_j\ge1\).

A random candidate covers a fixed structure with probability at least \(1/N_{\mathrm{cov}}\). A uniform \(X\in\binom{[\iota]}{\tilde t}\) has the right class counts with probability \(\prod_j\binom{q_j}{\tilde t_j}/\binom\iota {\tilde t}\), because the candidates with these counts are exactly the unions of a \(\tilde t_j\)-subset of each \(S_j\). Conditioned on that event, the sets \(X\cap S_j\) are independent and uniform among the \(\tilde t_j\)-subsets of \(S_j\), so by Lemma D.7.1 each is prefix-good with probability at least \(1/q_j\) (a class with \(q_j=0\) contributes the factor \(1\)). In particular \(N_{\mathrm{cov}}\ge\binom\iota {\tilde t}/\prod_j\binom{q_j}{\tilde t_j}\ge1\).

Size. Take \(N_{\mathrm{cand}}=\lfloor N_{\mathrm{cov}}\ln|\mathcal O|\rfloor+1\) independent uniform candidates. A fixed structure is uncovered by all of them with probability at most \((1-1/N_{\mathrm{cov}})^{N_{\mathrm{cand}}}<e^{-N_{\mathrm{cand}}/N_{\mathrm{cov}}}<1/|\mathcal O|\), so by the union bound some choice covers all structures, and the minimum cover has at most \(N_{\mathrm{cand}}\le N_{\mathrm{cov}}(1+\ln|\mathcal O|)\) members. A candidate covers at most \(|\mathcal O|\) structures, so greedy Set Cover returns a cover \(\mathcal B\) with \(|\mathcal B|\le(1+\ln|\mathcal O|)N_{\mathrm{cov}}(1+\ln|\mathcal O|)\le N_{\mathrm{cov}}(1+\ln\iota!)^2\).

Balance. Given disjoint \(S_j\subseteq[\iota]\) with \(|S_j|=q_j\) and \(w\ge0\), order each \(S_j\) by non-increasing \(w\) (ties broken arbitrarily), take a member of \(\mathcal B\) covering this ordered structure, and apply Lemma D.7.2 to each class \(j\) with \(q_j\ge1\): it gives \(w(X\cap S_j)\ge(\tilde t_j/q_j)w(S_j)\), simultaneously for all classes.

Time. The covering relation has \(|\mathcal O|\le\iota!\) elements and at most \(2^\iota\) candidate sets, and both it and the greedy computation take time polynomial in \(\iota!\,2^\iota\).  ∎

The block targets are distributed over the blocks by systematic rounding, which keeps every block target within one of its proportional share and has mean equal to that share. Formally, let \(p_1,\dots,p_b\ge0\) be integers with \(\sum_ip_i=k\ge1\), let \(t\in\{0,\dots,k\}\) and \(\delta=t/k\), and put \(p_{\le i}=p_1+\dots+p_i\) (\(p_{\le0}=0\)). For an offset \(\upsilon\in[0,1)\), the systematic rounding of \((p_i)\) with target \(t\) is \[t_i(\upsilon)=\lfloor\delta p_{\le i}+\upsilon\rfloor-\lfloor\delta p_{\le i-1}+\upsilon\rfloor\qquad(1\le i\le b).\]

Lemma D.7.4 (Systematic rounding).  In the setting above, for every \(\upsilon\in[0,1)\): \(t_i(\upsilon)\in\{\lfloor\delta p_i\rfloor,\lceil\delta p_i\rceil\}\), \(0\le t_i(\upsilon)\le p_i\) and \(\sum_it_i(\upsilon)=t\). For \(\upsilon\) uniform in \([0,1)\), \(\mathbb{E}\,t_i(\upsilon)=\delta p_i\).

Proof. For real \(x\) and \(x'\ge0\), \(\lfloor x+x'\rfloor-\lfloor x\rfloor\in\{\lfloor x'\rfloor,\lceil x'\rceil\}\); and \(0\le\lfloor\delta p_i\rfloor\le\lceil\delta p_i\rceil\le p_i\) since \(0\le\delta\le1\). The sum telescopes to \(\lfloor t+\upsilon\rfloor-\lfloor\upsilon\rfloor=t\). Finally, writing \(\{x\}\) for the fractional part, \(\mathbb{E}\lfloor x+\upsilon\rfloor=\lfloor x\rfloor+\Pr[\upsilon\ge1-\{x\}]=\lfloor x\rfloor+\{x\}=x\), so \(\mathbb{E}\,t_i(\upsilon)=\delta p_{\le i}-\delta p_{\le i-1}=\delta p_i\).  ∎

Two entropy facts control the size of the product construction.

Lemma D.7.5 (Entropy facts).  The binary entropy function \(H\) has the following properties.

(E1)  \(|H(x)-H(x')|\le H(|x-x'|)\) for \(x,x'\in[0,1]\).

(E2)  For an integer \(p\ge1\), a real \(\delta\in[0,1]\) and an integer \(q\) with \(|q-\delta p|<1\) and \(0\le q\le p\): \(pH(q/p)\ge pH(\delta)-\log(ep)\).

Proof. (E1) We may assume \(x\le x'\) and put \(h=x'-x\). Since \(H\) is concave, \(y\mapsto H(y+h)-H(y)\) is non-increasing on \([0,1-h]\), so \(H(x')-H(x)\le H(h)-H(0)=H(h)\). Applying this to \(1-x'\le1-x\) and using \(H(y)=H(1-y)\) gives \(H(x)-H(x')\le H(h)\).

(E2) For \(p=1\), \(|H(q)-H(\delta)|\le1\le\log e\). Let \(p\ge2\). By (E1), \(H(\delta)-H(q/p)\le H(|q/p-\delta|)\), and \(|q/p-\delta|<1/p\le\frac12\), so by monotonicity of \(H\) on \([0,\frac12]\) this is at most \(H(1/p)\). Finally \(pH(1/p)=\log p+(p-1)\log\frac p{p-1}\le\log p+(p-1)\frac{\log e}{p-1}=\log(ep)\), using \(\ln(1+x)\le x\).  ∎

The construction.

Let \(\iota\ge2\) be a block length. Partition \(U=[n]\) into \(b=\lceil n/\iota\rceil\) consecutive blocks \(U_1,\dots,U_b\), all of size \(n_i=\iota\) except possibly the last, and put \(\delta_j=t_j/k_j\) (\(\delta_j=0\) if \(k_j=0\)). The family \(\mathcal B\) consists of all sets \(X=\bigcup_i\mathrm{ord}_i(Y_i)\) where:

(1)  \((p_{ij})\) is a nonnegative integer \(b\times \mathsf r\) matrix with \(\sum_jp_{ij}\le n_i\) for every \(i\) and \(\sum_ip_{ij}=k_j\) for every \(j\);

(2)  \((t_{ij})\) has \(t_{ij}\in\{\lfloor\delta_jp_{ij}\rfloor,\lceil\delta_jp_{ij}\rceil\}\) and \(\sum_it_{ij}=t_j\) for every \(j\);

(3)  \(Y_i\) ranges over the block family \(\mathcal B_i=\mathcal B(n_i;(p_{ij})_j;(t_{ij})_j)\) of Lemma D.7.3 on \([n_i]\);

(4)  \(\mathrm{ord}_i\colon[n_i]\to U_i\) is the order-preserving bijection.

The matrix \((p_{ij})\) is a guess of how the unknown solution classes meet the blocks, and \((t_{ij})\) distributes the targets proportionally.

Proof of Theorem 9.6. Balance passes from the blocks to the whole universe through the rounding lemma; the size bound compares the product of the block bounds with the global binomial quotient, through an injection for the numerator and the entropy facts for the denominator; time and space follow from the order of enumeration.

Balance. Let \((S_j)\) and \(w\) be given, and put \(p_{ij}=|S_j\cap U_i|\); this matrix satisfies (1). Fix a class \(j\) with \(k_j\ge1\), and let \(t_{ij}(\upsilon_j)\) be the systematic rounding of \((p_{ij})_i\) with target \(t_j\) (Lemma D.7.4). The quantity \[\sum_{i:\,p_{ij}\ge1}\frac{t_{ij}(\upsilon_j)}{p_{ij}}\,w(S_j\cap U_i)\] has expectation \(\delta_jw(S_j)\) for uniform \(\upsilon_j\) (blocks with \(p_{ij}=0\) contain no element of \(S_j\)), so some \(\upsilon_j\) makes it at least \(\delta_jw(S_j)\). For a class with \(k_j=0\) all \(t_{ij}=0\). By Lemma D.7.4 the resulting matrix \((t_{ij})\) satisfies (2); the rounding parameters of different classes are chosen independently, and every combination occurs in (2). By Lemma D.7.3, applied in block \(i\) to the sets \(\mathrm{ord}_i^{-1}(S_j\cap U_i)\) and the weight \(w\circ\mathrm{ord}_i\), the family \(\mathcal B_i\) contains \(Y_i\) with \(|Y_i\cap\mathrm{ord}_i^{-1}(S_j)|=t_{ij}\), \(Y_i\subseteq\bigcup_j\mathrm{ord}_i^{-1}(S_j)\) and class-\(j\) weight at least \((t_{ij}/p_{ij})w(S_j\cap U_i)\), simultaneously for all \(j\) with \(p_{ij}\ge1\). The union \(X=\bigcup_i\mathrm{ord}_i(Y_i)\) lies in \(\mathcal B\), satisfies \(X\subseteq\bigcup_jS_j\) and \(|X\cap S_j|=\sum_it_{ij}=t_j\), and \(w(X\cap S_j)\ge\delta_jw(S_j)\) for every \(j\) with \(k_j\ge1\).

Size. Write \(t_i=\sum_jt_{ij}\), so that \(\sum_it_i=t\). The number of listed tuples is at most \(\#(p_{ij})\cdot\#(t_{ij})\cdot\max\prod_i|\mathcal B_i|\), where \(\#(p_{ij})\le(\iota+1)^{\mathsf rb}\) (each entry lies in \(\{0,\dots,\iota\}\)) and \(\#(t_{ij})\le2^{\mathsf rb}\). By Lemma D.7.3, \(|\mathcal B_i|\le\iota^{4}N_{\mathrm{cov},i}\), because \(1+\ln n_i!\le1+\iota\ln\iota\le1+\iota(\iota-1)\le\iota^2\). Also \(N_{\mathrm{cov},i}\le\iota^{\mathsf r}\binom{n_i}{t_i}/\prod_j\binom{p_{ij}}{t_{ij}}\). We bound the two products over \(i\).

• Numerator. \(\prod_i\binom{n_i}{t_i}\le\binom nt\), because choosing \(t_i\) elements in each block \(U_i\) gives distinct \(t\)-subsets of \(U\).

• Denominator. For \(p_{ij}\ge1\) we have \(|t_{ij}-\delta_jp_{ij}|<1\), so (1) and (E2) give \[\log\binom{p_{ij}}{t_{ij}}\ge p_{ij}H(\delta_j)-\log(ep_{ij})-\log(p_{ij}+1)\ge p_{ij}H(\delta_j)-\log\bigl(e\iota(\iota+1)\bigr),\] and the inequality between the outer terms also holds for \(p_{ij}=0\). Summing over \(i\) and using \(\sum_ip_{ij}H(\delta_j)=k_jH(\delta_j)\ge\log\binom{k_j}{t_j}\) (by (1)) gives \(\prod_{i,j}\binom{p_{ij}}{t_{ij}}\ge\prod_j\binom{k_j}{t_j}\cdot(e\iota(\iota+1))^{-\mathsf rb}\).

Altogether \[|\mathcal B|\le\frac{\binom nt}{\prod_j\binom{k_j}{t_j}} \bigl[2(\iota+1)\iota\cdot e\iota(\iota+1)\bigr]^{\mathsf rb}\iota^{4b} \le\frac{\binom nt}{\prod_j\binom{k_j}{t_j}}\,\iota^{12\mathsf rb},\] because \(2e\iota^{2}(\iota+1)^{2}\le\iota^{8}\) for \(\iota\ge2\) (at \(\iota=2\) the left side is \(72e<256\), and the ratio of the two sides decreases in \(\iota\)). Finally \(b\le n/\iota+1\). For \(n\ge16\) we have \(\iota\ge2\), and \(\iota\ge\log n/\log\log n-1\ge\frac12\log n/\log\log n\) once \(\log n/\log\log n\ge2\), which holds for \(n\ge16\); hence \(\log\iota^{12\mathsf r(n/\iota+1)}\le12\mathsf r(n/\iota+1)\log\log n=O(\mathsf rn(\log\log n)^2/\log n)\).

Time and space. Only two block sizes occur, so there are at most \(2(\iota+1)^{\mathsf r}2^{\mathsf r}\) distinct block parameters \((n_i;(p_{ij})_j;(t_{ij})_j)\), which is polynomial in \(n\) for fixed \(\mathsf r\). Each block family takes time polynomial in \(\iota!\,2^{\iota}\le n^2\), because \(\iota!\le\iota^{\iota}=2^{\iota\log\iota}\le2^{(\log n/\log\log n)\log\log n}=n\) and \(2^\iota\le n\); all block families are computed once and stored in polynomial space. The matrices \((p_{ij})\) are enumerated by depth-first search over the blocks, extending a partial choice only if the remaining column sums fit into the remaining blocks, that is, \(\sum_j(\text{remaining }k_j)\le\sum_{i'>i}n_{i'}\) after block \(i\); this is the exact feasibility condition, because any distribution of the remaining demand that respects the block capacities is admissible. The matrices \((t_{ij})\) are enumerated by depth-first search, keeping each remaining column sum between the sums of the remaining floors and ceilings, which is again exact. The product over blocks is enumerated by an odometer; every block family is nonempty, because for its parameters at least one ordered structure exists. So every leaf of the search yields a listed set, the time is \(n^{O(1)}\) per listed set plus \(n^{O(1)}\), and only the search state is kept.  ∎

Remark D.7.6 (Comparison with the construction of Fomin, Gaspers, Lokshtanov and Saurabh).  Compared with (Fomin et al. 2019), the construction of Theorem 9.6 uses no hashing: a fixed partition into blocks suffices because the block targets follow the block counts: the numerator of the size bound is at most \(\binom nt\) because distinct choices in the blocks give distinct \(t\)-sets, and the denominator loses only a factor \((e\iota(\iota+1))^{\mathsf r}=2^{O(\mathsf r\log\iota)}\) per block. Small blocks make the greedy step take polynomial time and space, and hence the whole construction runs in polynomial space. In exchange, the size bound has a larger subexponential factor, \(2^{O(n(\log\log n)^{2}/\log n)}\).

The subexponential factor.

With the explicit constants of Theorem 9.6, the factor by which the balanced families multiply the running-time bound of Theorem 10.4 is, for the two-class families of Algorithm 9.9 (\(\mathsf r=2\)), at most \(\iota^{24(n/\iota+1)}\) with \(\iota=\lfloor\log n/\log\log n\rfloor\), about \(2^{12718}\) at \(n=1000\) against \(1.7361^{1000}\approx2^{796}\). The product of this factor and \(1.7361^n\) is below the trivial \(2^n\) if and only if \(\iota\ge1203\), that is, if and only if \(\log n\ge16895.338\ldots\). These are upper bounds, not measured family sizes; for \(n\le20\) and block lengths \(4\) to \(6\) the actual product families contain 80–90% of all \(t\)-sets (Appendix F).

D.7.1 Trees across block partitions

The product construction of Theorem 9.6 handles conditions that are separable over a fixed partition of the universe into blocks. The analyses of RS-\(\theta\) and RSC rest on tests along an unknown spanning tree of the solution, and the following count shows that no family of \(2^{O(n)}\) fixed partitions into blocks of size \(n^{o(1)}\) avoids cutting almost all edges of some such tree. This is an obstacle for one construction strategy, not an impossibility result.

Proposition D.7.7 (Trees across block partitions).  Let \(S\) be a set of \(k\) elements, let \(c\ge0\) and \(B\ge1\), and let \(\mathcal P\) be a family of at most \(2^{ck}\) partitions of \(S\) into blocks of size at most \(B\). If \(\varepsilon\in(0,1]\) satisfies \(\varepsilon\log k>c+\log(B+1)+2+3(\log k)/k\), then some tree \(T\) with vertex set \(S\) has, for every partition in \(\mathcal P\), at least \((1-\varepsilon)k\) edges whose ends lie in different blocks. In particular, if \(B=k^{o(1)}\), then for every fixed \(\varepsilon>0\) and all large \(k\) some tree has \((1-\varepsilon)k\) such edges in every partition of the family.

Proof. Fix a partition in \(\mathcal P\). A tree \(T\) on \(S\) is determined by its restrictions to the blocks, which are forests, and by the set of its edges between blocks. A forest on \(b\) labeled vertices becomes a tree on \(b+1\) vertices when a new vertex is joined to one vertex of each component, and the forest is recovered by deleting the new vertex; so by Cayley’s formula there are at most \((b+1)^{b-1}\le(B+1)^{b}\) forests on a block of size \(b\), and at most \((B+1)^{k}\) choices of the restrictions in all. If \(m\) edges lie between blocks, they form an \(m\)-subset of the fewer than \(k^{2}\) pairs of elements in different blocks, which gives at most \(\binom{k^2}{m}\le(ek^2/m)^{m}\) choices for \(m\ge1\), a bound that is nondecreasing in \(m\) for \(m\le k^{2}\). For \(\varepsilon=1\) there is nothing to prove, so let \(\varepsilon<1\) and \(\alpha=1-\varepsilon\in(0,1)\). Then at most \(k(B+1)^{k}(ek/\alpha)^{\alpha k}\) trees have fewer than \(\alpha k\) edges between blocks of this partition (there are at most \(k\) values of \(m<\alpha k\)), and since \(\alpha\log(e/\alpha)\le\log e+\max_{0<\alpha\le1}\alpha\log(1/\alpha)<2\), this number is at most \(k(B+1)^{k}4^{k}k^{\alpha k}\). Over the family, at most \(2^{ck}k(B+1)^{k}4^{k}k^{(1-\varepsilon)k}=k^{k-2}\cdot2^{k(c+\log(B+1)+2)+3\log k-\varepsilon k\log k}\) trees have fewer than \((1-\varepsilon)k\) edges between blocks of some partition, and this is less than \(k^{k-2}\) because the exponent of \(2\) is negative by the hypothesis on \(\varepsilon\). By Cayley’s formula there are \(k^{k-2}\) trees on \(S\), so some tree has at least \((1-\varepsilon)k\) edges between blocks of every partition in \(\mathcal P\). The last sentence holds because \(c+\log(B+1)+2+3(\log k)/k=o(\log k)\) when \(B=k^{o(1)}\).  ∎

For a solution \(S\) of linear size \(k=\Theta(n)\), a family of \(2^{O(n)}\) partitions of the universe into blocks of size \(n^{o(1)}\) induces on \(S\) a family as in Proposition D.7.7, so some tree \(T\) on \(S\) has \((1-o(1))k\) edges between blocks of every partition of the family; on an instance in which \(S\) induces exactly the tree \(T\), these are edges of the only spanning tree of the solution.

D.8 The deterministic algorithms

This subsection proves the statements of Sections 9.4 and 10, with the notation of those sections.

D.8.1 Exponential moments of integer affine certificates

Restatement of Lemma 9.7.

Proof. The function \(\mathrm{cv}(x)=\ln(1+y2^{x})\) is convex. Let \(\mathrm{chord}\) be the affine function that agrees with \(\mathrm{cv}\) at \(a\) and \(a+1\). A convex function lies on or above its chord outside the chord’s interval, so \(\mathrm{cv}(v_u)\ge \mathrm{chord}(v_u)\) for every \(u\). Let \(w^*\) have \(k_a\) entries \(a\) and \(k_{a+1}\) entries \(a+1\); its total is \(ak+k_{a+1}=\Xi\). Since \(\mathrm{cv}=\mathrm{chord}\) at \(a\) and \(a+1\), and \(\mathrm{chord}\) is affine with \(\sum_uv_u=\sum_uw^*_u\), \[\sum_u\mathrm{cv}(v_u)\ge\sum_u\mathrm{chord}(v_u)=\sum_u\mathrm{chord}(w^*_u)=\sum_u\mathrm{cv}(w^*_u)=\ln \mathsf P^*(y).\] The second claim is Jensen’s inequality for \(\mathrm{cv}\).  ∎

Restatement of Proposition 9.8.

Proof. For a profile \(w\) let \(W^w_S(y)=\sum_t\mu^w_t(S)y^t=2^{-\alpha}\prod_{u\in S}(1+y2^{w(u)})\) be the generating function of the surrogate exponent. Lemma 3.9, whose proof uses only \(\mu_t(S)>0\) for \(0\le t\le k\) and \(W_S(y)=\sum_t\mu_t(S)y^t\), gives \(\mathsf R(w)\le\inf_{y\ge0}(1+y)^n/W^w_S(y)\le(n+1)\mathsf R(w)\). By the second claim of Lemma 9.7, \(W^w_S(y)\ge W^{\bar w}_S(y)\) for every \(y>0\), so \(\mathsf R(w)\le\inf_{y\ge0}(1+y)^n/W^w_S(y)\le\inf_{y\ge0}(1+y)^n/W^{\bar w}_S(y)\le(n+1)\mathsf R(\bar w)\). If \(w\) has integer values, it has no entry in \((a,a+1)\), and the first claim of Lemma 9.7 gives \(\mathsf R(w)\le(n+1)\mathsf R(w^*)\) in the same way. Finally, for the uniform profile \(\mu^{\bar w}_t(S)=\binom kt2^{-\alpha+t\Xi/k}\), which gives the formula for \(\mathsf R(\bar w)\).  ∎

Restatement of Theorem 9.10.

Proof. The early answer. If \(\alpha<\Xi\), no solution has an integer certificate: for \(X=S\) it would give \(0\le g(I,S,|S|)\le\alpha-\Xi<0\). So answering no is correct. From now on \(\alpha\ge\Xi\).

Number of guesses. There are at most \(k(\Xi+1)\le N^{O(1)}\) guesses, and for each the choice in step (2) ranges over \(O(k^2)\) pairs and takes polynomial time: the logarithm of each candidate is a rational plus logarithms of integers, so approximations to within \(\frac14\) are computable in time \(N^{O(1)}\), and a minimizer of the approximations is within a factor \(2^{1/2}\le2\) of the minimum.

Soundness. The algorithm answers yes only when \(\mathcal{A}\) accepts some \((I,X,k)\), which means that \(X\cup Y\in\mathcal{F}_I\) for some \(Y\) with \(|Y|\le k-t\), a solution of size at most \(k\).

Completeness. Let \(w\) be an integer certificate for \(S\). Put \(S^-=\{u\in S:w(u)\le a\}\) and \(S^+=S\setminus S^-\); as the weights are integers, every \(u\in S^+\) has \(w(u)\ge a+1\). Then \(S^-\ne\emptyset\), since otherwise \(\Xi\ge(a+1)k>\Xi\). The pair \((k^-,\Xi^-)=(|S^-|,w(S^-))\) satisfies \(1\le k^-\le k\) and \(\Xi^-\le ak^-\); moreover \(\Xi^+=w(S^+)\ge(a+1)k^+\), and if \(k^+=0\) then \(S^+=\emptyset\) and \(\Xi^+=0\). So this guess is tried. For it, the balanced family of step (3), applied to the classes \(S^-\), \(S^+\) and the weight \(w\), contains some \(X\subseteq S\) with \(|X\cap S^-|=t^-\), \(|X\cap S^+|=t^+\) and \(w(X)\ge(t^-/k^-)\Xi^-+(t^+/k^+)\Xi^+=t^-\bar w^-+t^+\bar w^+\) (the second term is absent if \(k^+=0\)). As \(t\ge1\), \(X\) is nonempty, so \(g(I,X,k)\le\alpha-w(X)\le\theta\), and \(\mathcal{A}\) is run on \((I,X,k)\) and accepts, with witness \(S\setminus X\).

The threshold is nonnegative. For every guess, \(\bar w^-,\bar w^+\ge0\), so \(t^-\bar w^-+t^+\bar w^+\le k^-\bar w^-+k^+\bar w^+=\Xi^-+\Xi^+=\Xi\le\alpha\). (For \(k^+=0\) this uses \(\Xi^-=\Xi\), which is the condition \(\Xi^+=0\).) Hence \(\theta\ge0\), and each listed set takes time at most \(N^{O(1)}+2^{\theta}N^{O(1)}\le2^{\theta}N^{O(1)}\).

Time. Fix a guess, and let \(v\in\mathbb{R}^k\) have \(k^-\) entries \(\bar w^-\) and \(k^+\) entries \(\bar w^+\). Its total is \(\Xi^-+\Xi^+=\Xi\), and it has no entry in \((a,a+1)\), because \(\bar w^-\le a\) and \(\bar w^+\ge a+1\). For \(1\le t\le k\) put \[\mathsf P_t=\sum_{t^-+t^+=t}\binom{k^-}{t^-}\binom{k^+}{t^+}2^{t^-\bar w^-+t^+\bar w^+},\] a positive sum of at most \(t+1\) terms. Its largest term is at least \(\mathsf P_t/(t+1)\), so by the choice in step (2), which is within a factor \(2\) of the minimum, and by Theorem 9.6, the running time of the guess is at most \[2^{O(n(\log\log n)^{2}/\log n)}N^{O(1)}\min_{1\le t\le k}(t+1)\binom nt\frac{2^{\alpha}}{\mathsf P_t}.\] Expanding the product over the entries of \(v\) by subsets and grouping the subsets by their numbers of light and heavy entries gives, for every \(y>0\), \[\sum_{t=1}^{k}\mathsf P_ty^t=\prod_u\bigl(1+y2^{v_u}\bigr)-1\le\max_{1\le t\le k}\frac{\mathsf P_t}{\binom nt}\sum_{t=1}^{k}\binom nty^t \le\max_{1\le t\le k}\frac{\mathsf P_t}{\binom nt}\bigl((1+y)^{n}-1\bigr),\] so \(\min_{1\le t\le k}\binom nt/\mathsf P_t\le((1+y)^{n}-1)/(\prod_u(1+y2^{v_u})-1)\). Lemma 9.7 bounds the product from below by \(\mathsf P^*(y)\), and \(\mathsf P^*(y)>1\) for \(y>0\). As this holds for every \(y>0\), and \(t+1\le n+1\), the running time of the guess is at most \(2^{O(n(\log\log n)^{2}/\log n)}N^{O(1)}\inf_{y>0}((1+y)^{n}-1)2^{\alpha}/(\mathsf P^*(y)-1)\). Multiplying by the polynomial number of guesses, and adding the polynomial time of the early answer and of the choices in steps (1) and (2), which the maximum with \(1\) absorbs, gives the time bound.

Space. The guesses are run one after another, each in the space of Theorem 9.6 plus that of \(\mathcal{A}\).  ∎

D.8.2 Deterministic Connected Dominating Set

Restatement of Lemma 10.1.

Proof. For fixed \(y\) the function \(\chi_y\) is affine in \(u\) on each interval \([\frac1{a+1},\frac1a]\), so it suffices to check endpoints. The proof has an algebraic part, on one interval, where a single \(y\) gives equality at both endpoints, and a rational part, on five intervals, where the endpoint inequalities are inequalities between rational numbers.

Reduction to endpoints. On each interval \(u\in[\frac1{a+1},\frac1a]\) we use the formula with this value of \(a\). For \(u\) in the interior or at the right endpoint this is \(\lfloor1/u\rfloor\); at the left endpoint \(u=\frac1{a+1}\) we have \(\lfloor1/u\rfloor=a+1\), but both formulas give the same value there, namely \(\log(1+y)+1-\frac1{a+1}\log(1+y2^{a+1})\), since the coefficients are \((0,\frac1{a+1})\) for \(a\) and \((\frac1{a+1},0)\) for \(a+1\). For fixed \(y\) and \(a\) the two coefficients \((a+1)u-1\) and \(1-au\) are affine in \(u\), so \(\chi_y\) is affine on the interval and attains its maximum at an endpoint. The intervals that meet \([0.15,0.85]\) are those with \(a=1,\dots,6\).

The interval \(a=2\), \(u\in[\frac13,\frac12]\) (algebraic). Take \(y_0=(1+\sqrt5)/8\). Then \(1+4y_0=\frac{3+\sqrt5}2=\left(\frac{1+\sqrt5}2\right)^2\), \(1+8y_0=2+\sqrt5=\left(\frac{1+\sqrt5}2\right)^3\) and \(2(1+y_0)=(9+\sqrt5)/4\). At \(u=\frac12\) the coefficients are \(\frac12\) and \(0\), so \(2^{\chi_{y_0}(1/2)}=2(1+y_0)/(1+4y_0)^{1/2}\). At \(u=\frac13\) they are \(0\) and \(\frac13\), so \(2^{\chi_{y_0}(1/3)}=2(1+y_0)/(1+8y_0)^{1/3}\). Both equal \[\frac{9+\sqrt5}{4}\cdot\frac{2}{1+\sqrt5}=\frac{9+\sqrt5}{2(1+\sqrt5)}=\frac{(9+\sqrt5)(\sqrt5-1)}{8}=\frac{8\sqrt5-4}8=\sqrt5-\frac12 .\] Since \(\chi_{y_0}\) is affine on the interval, \(\chi_{y_0}(u)=\log(\sqrt5-\frac12)\) for every \(u\in[\frac13,\frac12]\).

The other intervals (rational). Use \(y=\frac12\) for \(a=1\) on \([\frac12,\frac{17}{20}]\), \(y=\frac5{16}\) for \(a=3\) on \([\frac14,\frac13]\), \(y=\frac14\) for \(a=4\) on \([\frac15,\frac14]\), \(y=\frac{27}{128}\) for \(a=5\) on \([\frac16,\frac15]\), and \(y=\frac{29}{160}\) for \(a=6\) on \([\frac3{20},\frac16]\). At each of the ten endpoints \(u\), both coefficients are nonnegative rationals; with \(h\) their least common denominator, the inequality \(\chi_y(u)\le\log\frac{1736}{1000}\) is equivalent to \[\bigl(2(1+y)\bigr)^{h}\ \le\ \Bigl(\frac{1736}{1000}\Bigr)^{h}\,\bigl(1+y2^{a}\bigr)^{h((a+1)u-1)}\bigl(1+y2^{a+1}\bigr)^{h(1-au)},\] an inequality between rational numbers with nonnegative integer exponents. It holds at all ten endpoints, by an exact evaluation (Appendix F.5). The largest endpoint value of \(2^{\chi_y(u)}\) on these intervals is \(\sqrt3=1.7320\ldots\), at \(u=\frac12\) for \(a=1\). Since \(\frac{1736}{1000}<\sqrt5-\frac12\), the lemma holds on these intervals. As the five intervals and \([\frac13,\frac12]\) cover \([0.15,0.85]\), the lemma follows.  ∎

Restatement of Lemma 10.2.

Proof. Write \(u=s/\Xi\) and \(a=\lfloor1/u\rfloor=\lfloor\Xi/k\rfloor\). In the notation of Theorem 9.10, \(k_{a+1}=\Xi-as=\Xi(1-au)\) and \(k_a=s-k_{a+1}=\Xi((a+1)u-1)\), so \[\log \mathsf P^*(y)=\Xi\,\mathsf p_y,\qquad \mathsf p_y=\bigl((a+1)u-1\bigr)\log\bigl(1+y2^{a}\bigr)+(1-au)\log\bigl(1+y2^{a+1}\bigr).\] The two coefficients of \(\mathsf p_y\) are nonnegative and sum to \(u\le1\), so \(0\le \mathsf p_y\le\log(1+y2^{a+1})\). If \(\mathsf P^*(y)\ge2\), then \(\mathsf P^*(y)-1\ge \mathsf P^*(y)/2\), and, as \((1+y)^{n}-1\le(1+y)^{n}\) and \(\alpha+1\le n+1\), \[\frac1n\log\frac{\bigl((1+y)^{n}-1\bigr)\,2^{\alpha}}{\mathsf P^*(y)-1} \le\frac1n\bigl(n\log(1+y)+\alpha+1-\Xi\mathsf p_y\bigr)\le\chi_y(u)+\frac{1+2\mathsf p_y}n ,\] using \(\chi_y(u)=\log(1+y)+1-\mathsf p_y\) in the notation of Lemma 10.1, \(\alpha+1=\Xi+2\le n+1\) and \(\Xi\mathsf p_y\ge(n-2)\mathsf p_y\).

We check the range of \(u\). Since \(\Xi\le n-1<n\) and \(n\ge34\), \(u>0.15n/\Xi>0.15\), and \(u<0.8n/\Xi\le0.8n/(n-2)\le0.85\), where the last inequality is equivalent to \(n\ge34\). So \(u\in[0.15,0.85]\), and \(1\le a\le6\) because \(1/u\in[1/0.85,1/0.15]\). Lemma 10.1 gives, for this \(u\), some \(y\) from its list with \(\chi_y(u)\le\log(\sqrt5-\frac12)\). Every listed \(y\) is at least \(\frac{29}{160}\), so \[\mathsf P^*(y)\ge1+(k_a+k_{a+1})\,y2^{a}=1+sy2^{a}\ge1+0.15\cdot34\cdot2\cdot\tfrac{29}{160}>2 .\] Moreover \(y\le\frac12\) and \(a\le6\), so \(\mathsf p_y\le\log(1+2^{6})\) is bounded by a constant. Hence Theorem 9.10, evaluated at this \(y\), bounds the time of the run by \[N^{O(1)}\,2^{O(n(\log\log n)^{2}/\log n)}\max\bigl(1,2^{n\chi_y(u)+1+2\mathsf p_y}\bigr) \le N^{O(1)}\,2^{O(n(\log\log n)^{2}/\log n)}\,\bigl(\sqrt5-\tfrac12\bigr)^{n},\] where the last step uses \(\chi_y(u)\le\log(\sqrt5-\frac12)\), \(\mathsf p_y\ge0\) and \(\sqrt5-\frac12>1\).  ∎

Restatement of Lemma 10.3.

Proof. The enumeration is deterministic, uses polynomial space, since membership in \(\mathcal{F}_I\) is tested in polynomial time (Definition 3.1), and answers yes exactly when \(\mathcal{F}_I\) has a solution of size \(s\). It takes time \(\binom nsN^{O(1)}\), which is \(N^{O(1)}\) for \(n<34\) and otherwise at most \(2^{H(0.15)n}N^{O(1)}=O(1.5261^n)N^{O(1)}\) for \(s\le0.15n\) and \(2^{H(0.2)n}N^{O(1)}=O(1.6494^n)N^{O(1)}\) for \(s\ge0.8n\), by (1), since the binary entropy function \(H\) is increasing on \([0,\frac12]\) and symmetric about \(\frac12\); both bases are below \(\sqrt5-\frac12\). In the other case, \(\alpha\) is a rational and \(\Xi\) a nonnegative integer at most \(n\), both computable in polynomial time, and \(\alpha\ge\Xi\), so Algorithm 9.9 does not stop early. Theorem 9.10 gives that the run is deterministic, uses space \(N^{O(1)}\) plus the space of \(\mathcal{A}\), answers no if \(\mathcal{F}_I\) has no solution of size at most \(s\), and answers yes if a solution of size \(s\) has an integer certificate; Lemma 10.2 gives the time bound.  ∎

Restatement of Theorem 10.4.

Proof. For \(s=1,\dots,n\) run the procedure \(\mathrm{Det}(G,s)\) of Lemma 10.3 for the implicit set system \(\Phi_{\mathrm{CDS}}\), the extension algorithm \(\mathcal{A}_{\mathrm{St}}\) with its running-time exponent \(g(G,X,k)=|\mathrm{T}(J_X)|\), which is deterministic, uses polynomial space and has values in \(\{0,\dots,n\}\) by Corollary 5.8, and \((\alpha,\Xi)=(n-1,n-2)\); output the least \(s\) at which it answers yes. By Lemma 10.3, \(\mathrm{Det}(G,s)\) answers no if \(\gamma_{\mathrm{c}}(G)>s\). At \(s=\gamma_{\mathrm{c}}(G)\) a minimum connected dominating set \(S\) has size exactly \(s\), and, for \(n\ge2\), the weights \(w=d_F-1\) of (19) are an integer certificate for \(S\): they are nonnegative integers with \(w(S)=n-2=\Xi\) and \(g(G,X,s)=|\mathrm{T}(J_X)|\le\alpha-w(X)\) for every nonempty \(X\subseteq S\). So \(\mathrm{Det}(G,\gamma_{\mathrm{c}}(G))\) answers yes, and the deterministic form of Lemma 5.2 shows that the output is \(\gamma_{\mathrm{c}}(G)\), and that the same algorithm solves Max Leaf Spanning Tree. The time is \(n\) times the bound of Lemma 10.3, which is \(O(1.7361^n)\) because \(2^{O(n(\log\log n)^2/\log n)}=2^{o(n)}\) and \(\sqrt5-\frac12<1.7361\), and the space is polynomial.  ∎

D.9 Connected Feedback Vertex Set

This subsection proves the two hardness facts of Section 11.3.1 and constructs the extension algorithm of Lemma 11.7. We keep the notation of Section 11.3.

D.9.1 Hardness of the extension problem

Lemma D.9.1 (The extension problem is W[2]-hard).  Connected Feedback Vertex Set Extension, parameterized by the number \(k-|X|\) of further vertices, is W[2]-hard.

Proof. We reduce from Set Cover parameterized by the number of sets, which is W[2]-hard (Cygan et al. 2015). Let \((\mathcal U,\mathcal S,k)\) be an instance with ground set \(\mathcal U\), a family \(\mathcal S\) of subsets of \(\mathcal U\) and an integer \(k\). We may assume that \(\mathcal U\ne\emptyset\), that \(\mathcal S\neq\emptyset\) and that every element lies in some set, since otherwise the instance is trivial. Build a graph \(G\) with a vertex for every element, a vertex for every set and a root \(r\); join \(r\) to every set vertex, and join each set vertex to the elements it contains. Then \(G\) is connected. Put \(X=\{r\}\cup\mathcal U\) and take the size bound \(|X|+k\), so that \(k\) further vertices may be added.

The vertices outside \(X\) are the set vertices. Let \(Y\) be a set of set vertices. The graph \(G-(X\cup Y)\) consists of set vertices only, which are pairwise non-adjacent, so it is a forest. In \(G[X\cup Y]\) the vertices of \(X\) are pairwise non-adjacent and every vertex of \(Y\) is adjacent to \(r\). Hence \(G[X\cup Y]\) is connected if and only if every element vertex has a neighbor in \(Y\), that is, if and only if \(Y\) covers \(\mathcal U\). So \((G,X,|X|+k)\) is a yes-instance if and only if \(\mathcal U\) has a cover by at most \(k\) sets. The reduction is polynomial and keeps the parameter.  ∎

The next statement uses the Set Cover Conjecture of (Cygan et al. 2016): for every \(\varepsilon>0\) there is an integer \(h\) such that Set Cover with sets of size at most \(h\), on a universe \(\mathcal U\) of \(n_{\mathcal U}\) elements and with \(|\mathcal S|\) sets, cannot be solved in time \((2-\varepsilon)^{n_{\mathcal U}}(n_{\mathcal U}+|\mathcal S|)^{O(1)}\).

Proposition D.9.2 (Product-form running times cannot help).  Assume the Set Cover Conjecture. Let \(c,d\ge1\) be constants such that Connected Feedback Vertex Set Extension can be solved in time \(c^{k-|X|}d^{|X|}N^{O(1)}\) on inputs \((G,X,k)\). Then \(c\ge2\) and \(d\ge2\), and the running-time bound of Corollary 4.6 with \((b,c,d)=(1,c,d)\) has a base at least \(2\).

Proof. Suppose that \(d<2\). Put \(\varepsilon=(2-d)/2>0\) and fix an integer \(j\ge1\) with \(c^{1/j}d\le2-\varepsilon\); it exists because \(c^{1/j}\to1\) and \(d<2-\varepsilon\). Neither \(\varepsilon\) nor \(j\) depends on \(h\). Let \(h\) be arbitrary and let \((\mathcal U,\mathcal S,k)\) be an instance of Set Cover with sets of size at most \(h\) and \(n_{\mathcal U}=|\mathcal U|\); we may assume \(k\le n_{\mathcal U}\), since otherwise the answer is yes if and only if every element lies in some set.

1.  Padding. Let \(\hat k=\lceil k/j\rceil\) and \(p=j\hat k-k<j\). Add \(p\) new elements, each with its own singleton set. Every cover of the new instance contains the \(p\) singletons, because they are the only sets containing the new elements. Hence the new instance has a cover with at most \(j\hat k\) sets if and only if the old one has a cover with at most \(k\) sets.

2.  Unions. Replace the sets by all unions of at most \(j\) of them. There are at most \((|\mathcal S|+j+1)^{j}\) such unions, a polynomial for fixed \(j\). The padded instance has a cover with at most \(j\hat k\) sets if and only if the union instance has a cover with at most \(\hat k\) sets: group the sets of a cover into \(\hat k\) groups of at most \(j\), and conversely split each union.

3.  Reduction. Apply the reduction of Lemma D.9.1 to the union instance. It produces a forced set with \(|X|\le n_{\mathcal U}+j+1\) and \(\hat k\le n_{\mathcal U}/j+1\) further vertices, so the assumed algorithm decides it in time \[c^{\hat k}d^{\,n_{\mathcal U}+j+1}N^{O(1)}\le c\,d^{j+1}\bigl(c^{1/j}d\bigr)^{n_{\mathcal U}}N^{O(1)} \le(2-\varepsilon)^{n_{\mathcal U}}(n_{\mathcal U}+|\mathcal S|)^{O(1)},\] since \(N\) is polynomial in \(n_{\mathcal U}+|\mathcal S|\) for fixed \(j\).

This holds for every \(h\) with the same \(\varepsilon\), contradicting the conjecture. Hence \(d\ge2\).

Now suppose that \(c<2\). Put \(\varepsilon=(2-c)/2>0\) and fix an integer \(j\ge1\) with \(c^{1+1/j}\le2-\varepsilon\); it exists because \(c^{1/j}\to1\) and \(c<2-\varepsilon\). Let \(h\) be arbitrary, let \((\mathcal U,\mathcal S,k)\) be an instance of Set Cover with sets of size at most \(h\), \(n_{\mathcal U}=|\mathcal U|\ge1\) and \(k\le n_{\mathcal U}\), in which every element lies in some set (otherwise the answer is no), and apply the padding and union steps above. They yield an instance \((\mathcal U',\mathcal S',\hat k)\) with \(n_{\mathcal U}\le|\mathcal U'|<n_{\mathcal U}+j\) and \(\hat k\le n_{\mathcal U}/j+1\), in which every element lies in some set, and which has a cover with at most \(\hat k\) sets if and only if the original instance has a cover with at most \(k\) sets. Build the graph of the proof of Lemma D.9.1 for \((\mathcal U',\mathcal S')\), with a root \(r\), one vertex for every set of \(\mathcal S'\) and one vertex for every element of \(\mathcal U'\), and attach to \(r\) and to every element vertex \(v\) a private triangle: two new vertices \(v',v''\), adjacent to \(v\) and to each other. Call \(r\) and the element vertices anchors; as \(\mathcal U'\ne\emptyset\), there are at least two of them. Let \(G''\) be the resulting connected graph, put \(X=\emptyset\) and \(s=|\mathcal U'|+1+\hat k\). We claim that \(G''\) has a connected feedback vertex set of size at most \(s\) if and only if \(\mathcal U'\) has a cover by at most \(\hat k\) sets of \(\mathcal S'\).

If \(Y\) is such a cover, then \(A=\{r\}\cup\mathcal U'\cup Y\), viewed as a vertex set, is a connected feedback vertex set of size at most \(s\): \(G''[A]\) is connected, because every set vertex is adjacent to \(r\) and every element vertex is adjacent to a set vertex in \(Y\); and \(G''-A\) consists of isolated set vertices and the edges \(v'v''\), so it is a forest. Conversely, let \(A\) be a connected feedback vertex set of \(G''\) with \(|A|\le s\). Every anchor lies in \(A\): if an anchor \(v\) is not in \(A\), then \(v'\) or \(v''\) is, because \(A\) meets the triangle at \(v\); the neighbors of \(v'\) and of \(v''\) are \(v\) and each other, so the connectivity of \(G''[A]\) forces \(A\subseteq\{v',v''\}\), and then \(G''-A\) contains the triangle of another anchor, a contradiction. Removing all triangle vertices from \(A\) keeps it a connected feedback vertex set of no larger size: a triangle vertex is adjacent only to its anchor, which lies in \(A\), and to its twin, so it is an interior vertex of no path of \(G''[A]\) between other vertices, and returning \(v'\) and \(v''\) to \(G''-A\) adds at most the isolated edge \(v'v''\). So we may assume that \(A=\{r\}\cup\mathcal U'\cup Y\) for a set \(Y\) of at most \(s-|\mathcal U'|-1=\hat k\) set vertices, and the connectivity of \(G''[A]\) means that every element vertex has a neighbor in \(Y\), that is, that \(Y\) covers \(\mathcal U'\). This proves the claim.

The assumed algorithm decides \((G'',\emptyset,s)\) in time \[c^{s}N^{O(1)}\le c^{\,n_{\mathcal U}+j+n_{\mathcal U}/j+2}N^{O(1)}=c^{j+2}\bigl(c^{1+1/j}\bigr)^{n_{\mathcal U}}N^{O(1)} \le(2-\varepsilon)^{n_{\mathcal U}}(n_{\mathcal U}+|\mathcal S|)^{O(1)},\] again for every \(h\) with the same \(\varepsilon\), contradicting the conjecture. Hence \(c\ge2\).

With \(b=1\) the base of Corollary 4.6 is \(1+\min(d,c)(1-1/c)\). If \(d\ge c\), it equals \(c\ge2\). If \(d<c\), it equals \(1+d-d/c>1+d-1=d\ge2\).  ∎

Proposition D.9.2 concerns extension algorithms whose running-time bound depends only on \(k-|X|\) and \(|X|\) (the case \(b=1\)); it says nothing about bounds with a further factor \(b^{n-|X|}\), \(b>1\).

D.9.2 Contractions

The extension algorithm contracts the components of the sample, and its iterative compression runs along a sequence of edge contractions. This subsection collects the facts about contractions that the proofs use. For a graph \(\tilde G\) and an edge \(uv\in E(\tilde G)\), the graph \(\tilde G/uv\) is obtained by replacing \(u\) and \(v\) by a new vertex \(w\) adjacent to \(N_{\tilde G}(u)\cup N_{\tilde G}(v)\setminus\{u,v\}\); it is simple. For a set \(C\subseteq V(\tilde G)\) with \(\tilde G[C]\) connected, contracting \(C\) means replacing \(C\) by one vertex adjacent to \(N_{\tilde G}(C)\); it is the result of \(|C|-1\) edge contractions inside \(C\). We use the standard count: a graph with \(p\) vertices, \(e\) edges and \(q\) components satisfies \(e\ge p-q\), with equality if and only if it is a forest, since a connected graph on \(p_i\) vertices has at least \(p_i-1\) edges, with equality if and only if it is a tree. The lemma below records that an edge contraction preserves the number of components and the property of being a forest, and that contracting the sample does not change the extension problem.

Let \(G\) be connected, \(X\subseteq V(G)\) nonempty, and let \(X_1,\dots,X_q\) be the vertex sets of the components of \(G[X]\), so \(q=\operatorname{comp}(G[X])\). The contracted graph \(G_X\) is obtained from \(G\) by contracting each \(X_j\) to a new vertex \(x_j\). Its vertex set is \((V(G)\setminus X)\cup \hat X\) with \(\hat X=\{x_1,\dots,x_q\}\); we call the vertices of \(\hat X\) forced. Distinct components of \(G[X]\) are not adjacent, so \(\hat X\) is an independent set of \(G_X\).

Lemma D.9.3 (Contractions). 

(1)  Let \(\tilde G\) be a graph and \(uv\in E(\tilde G)\). Then \(\operatorname{comp}(\tilde G/uv)=\operatorname{comp}(\tilde G)\); in particular, \(\tilde G\) is connected if and only if \(\tilde G/uv\) is. If \(\tilde G\) is a forest, then \(\tilde G/uv\) is a forest.

(2)  With \(G\), \(X\), \(G_X\) and \(\hat X\) as above, \(G_X\) is connected, and for every \(Y\subseteq V(G)\setminus X\), the set \(X\cup Y\) is a connected feedback vertex set of \(G\) if and only if \(\hat X\cup Y\) is a connected feedback vertex set of \(G_X\).

Proof. (1) The map \(h\) that sends \(u\) and \(v\) to \(w\) and fixes the other vertices sends every edge of \(\tilde G\) other than \(uv\) to an edge of \(\tilde G/uv\), and every edge of \(\tilde G/uv\) is the image of an edge of \(\tilde G\); since \(h^{-1}(w)=\{u,v\}\) is connected in \(\tilde G\), \(h\) induces a bijection between the components. If \(\tilde G\) is a forest, then \(\tilde G/uv\) has one vertex fewer, at least one edge fewer and the same number of components; as \(|E(\tilde G)|=|V(\tilde G)|-\operatorname{comp}(\tilde G)\), the count \(e\ge p-q\) holds with equality for \(\tilde G/uv\), so it is a forest. (2) \(G_X\) arises from the connected graph \(G\) by edge contractions, so it is connected by (1). The graphs \(G-(X\cup Y)\) and \(G_X-(\hat X\cup Y)\) are identical: both have vertex set \(V(G)\setminus(X\cup Y)\), and contracting the sets \(X_j\) does not change the edges between vertices outside \(X\). The graph \(G_X[\hat X\cup Y]\) is obtained from \(G[X\cup Y]\) by contracting the connected sets \(X_1,\dots,X_q\), so by (1) one is connected if and only if the other is.  ∎

For the iterative compression we need contraction closure in the presence of forced vertices, which adapts the observation in the proof of Theorem B.3 of (Cygan et al. 2011). Let \(\tilde G\) be a graph with a forced set \(\hat X\), let \(uv\in E(\tilde G)\), let \(\tilde G_{uv}=\tilde G/uv\) with new vertex \(w\), and let the forced set \(\hat X_{uv}\) of \(\tilde G_{uv}\) be the image of \(\hat X\), so \(w\in \hat X_{uv}\) if and only if \(u\in \hat X\) or \(v\in \hat X\). The excess of a set \(A\) in a graph with forced set \(\hat X\) is \(|A\setminus \hat X|\).

Lemma D.9.4 (Contraction closure with forced vertices).  Let \(\tilde G\), \(\hat X\), \(uv\), \(\tilde G_{uv}\), \(w\) and \(\hat X_{uv}\) be as above, and let \(A\) be a connected feedback vertex set of \(\tilde G\) with \(\hat X\subseteq A\). Put \(A'=A\) if \(u,v\notin A\), and \(A'=(A\setminus\{u,v\})\cup\{w\}\) otherwise. Then \(A'\) is a connected feedback vertex set of \(\tilde G_{uv}\), \(\hat X_{uv}\subseteq A'\), and \(|A'\setminus \hat X_{uv}|\le|A\setminus \hat X|\).

Proof. Suppose first that \(u,v\notin A\). Then \(A'=A\), \(\tilde G_{uv}[A]=\tilde G[A]\), which is connected, and \(\tilde G_{uv}-A=(\tilde G-A)/uv\), which is a forest by Lemma D.9.3(1). Since \(w\notin A'\) and \(\hat X\subseteq A\) contains neither \(u\) nor \(v\), we have \(\hat X_{uv}=\hat X\), so \(\hat X_{uv}\subseteq A'\) and \(A'\setminus \hat X_{uv}=A\setminus \hat X\).

Suppose now that \(u\in A\) or \(v\in A\). Then \(\tilde G_{uv}-A'=\tilde G-(A\cup\{u,v\})\) is an induced subgraph of the forest \(\tilde G-A\). The set \(A\cup\{u,v\}\) induces a connected subgraph of \(\tilde G\), because \(\tilde G[A]\) is connected and \(uv\) is an edge with an end in \(A\); and \(\tilde G_{uv}[A']=\tilde G[A\cup\{u,v\}]/uv\) is connected by Lemma D.9.3(1). Every forced vertex of \(\tilde G_{uv}\) other than \(w\) lies in \(\hat X\setminus \{u,v\}\subseteq A\setminus\{u,v\}\subseteq A'\), and \(w\in A'\); so \(\hat X_{uv}\subseteq A'\). Finally, \(A'\setminus \hat X_{uv}\subseteq(A\setminus \hat X\setminus\{u,v\})\cup(\{w\}\setminus \hat X_{uv})\). If \(w\notin \hat X_{uv}\), then \(u,v\notin \hat X\), and at least one of them lies in \(A\setminus \hat X\), so the excess does not increase. If \(w\in \hat X_{uv}\), the excess is at most \(|A\setminus \hat X|\) directly.  ∎

D.9.3 Cut & Count with forced vertices

The counting step of the extension algorithm is the Cut & Count algorithm of (Cygan et al. 2011, 2022) for the constrained problem, run on a tree decomposition of the form “forest plus a fixed modulator,” as in their proof of the bound \(3^kn^{O(1)}\) for Connected Feedback Vertex Set. The journal version (Cygan et al. 2022) states these bounds but leaves their proofs to the extended arXiv version (Cygan et al. 2011) (see (Cygan et al. 2022, sec. 1.5)), so the numbers below that concern Connected Feedback Vertex Set refer to (Cygan et al. 2011); the corresponding results for Feedback Vertex Set are (Cygan et al. 2022 Theorems 4.6 and 5.1). Constrained Connected Feedback Vertex Set asks, given a graph \(\tilde G\) with \(\tilde n\) vertices, a nonempty set \(Y_{\mathrm{in}}\subseteq V(\tilde G)\) and an integer \(\eta\), whether \(\tilde G\) has a connected feedback vertex set \(Y\supseteq Y_{\mathrm{in}}\) with \(|Y|=\eta\). We use the following facts from these papers.

(CC1)  ((Cygan et al. 2011 Theorem A.13).) Given a tree decomposition of \(\tilde G\) of width \(w\), a Monte Carlo algorithm decides Constrained Connected Feedback Vertex Set in time \(4^w\tilde n^{O(1)}\). It never accepts a no-instance, and it rejects a yes-instance with probability at most \(1/2\).

(CC2)  (Proof of (Cygan et al. 2011 Theorem A.13).) The algorithm fixes a vertex \(r\in Y_{\mathrm{in}}\) and computes, modulo \(2\), the number of candidate solutions with a marked candidate forest (markers on side \(1\) of its cut), together with a consistent cut of the candidate forest and a consistent cut of the candidate solution that puts \(r\) on side \(1\). Its dynamic program gives every vertex of a bag one of four states, which we write \(\mathsf y_1,\mathsf y_2\) (in the solution, on side \(1\) or \(2\) of its cut) and \(\mathsf z_1,\mathsf z_2\) (in the forest, on side \(1\) or \(2\) of its cut; these are the states \(0_1,0_2,1_1,1_2\) of (Cygan et al. 2011)). A counted object gives a state \(\mathsf y_1\) or \(\mathsf y_2\) to every vertex of \(Y_{\mathrm{in}}\) and the state \(\mathsf y_1\) to \(r\), and no edge has the pair of states \(\{\mathsf y_1,\mathsf y_2\}\) or \(\{\mathsf z_1,\mathsf z_2\}\), because both cuts are consistent.

(CC3)  (Proof of (Cygan et al. 2022 Theorem 5.1) for Feedback Vertex Set, which the proof of Theorem B.3 of (Cygan et al. 2011) repeats for the connected problem.) Let \(\mathrm{Mod}\subseteq V(\tilde G)\) be such that \(\tilde G-\mathrm{Mod}\) is a forest, and use the tree decomposition obtained from a decomposition of width \(1\) of \(\tilde G-\mathrm{Mod}\) by adding \(\mathrm{Mod}\) to every bag. Below the bags that forget the vertices of \(\mathrm{Mod}\), each table entry for a coloring \(\varsigma\) of \(\mathrm{Mod}\) refers only to entries with the same coloring of \(\mathrm{Mod}\). So the count is the sum over the colorings \(\varsigma\colon\mathrm{Mod}\to\{\mathsf y_1,\mathsf y_2,\mathsf z_1,\mathsf z_2\}\) of the counts with \(\mathrm{Mod}\) colored by \(\varsigma\) (the “core evaluations”), and each of these is computed in time and space polynomial in \(\tilde n\).

By (CC2), only the following colorings of \(\mathrm{Mod}\) can have a nonzero count. A coloring \(\varsigma\colon \mathrm{Mod}\to\{\mathsf y_1,\mathsf y_2,\mathsf z_1,\mathsf z_2\}\) is reasonable if \(\varsigma(v)\in\{\mathsf y_1,\mathsf y_2\}\) for \(v\in Y_{\mathrm{in}}\cap \mathrm{Mod}\), \(\varsigma(r)=\mathsf y_1\) when \(r\in \mathrm{Mod}\), and no edge \(uv\) of \(\tilde G[\mathrm{Mod}]\) has \(\{\varsigma(u),\varsigma(v)\}=\{\mathsf y_1,\mathsf y_2\}\) or \(\{\varsigma(u),\varsigma(v)\}=\{\mathsf z_1,\mathsf z_2\}\). Summing the core evaluations over the reasonable colorings therefore gives the same parity as the algorithm of (CC1). The following lemma bounds the number of reasonable colorings for the modulators produced by iterative compression. It is the forced-vertex version of the count \(4^33^{k-1}\) in the proof of Theorem B.3 of (Cygan et al. 2011), in which only the vertex \(r\) and the at most two vertices outside the connected part of the modulator have four possible states.

Lemma D.9.5 (Number of reasonable colorings).  Let \(\tilde G\), \(Y_{\mathrm{in}}\) and \(r\) be as above. Let \(\mathrm{Mod}=C\cup C'\) with \(C\cap C'=\emptyset\), \(\tilde G[C]\) connected and \(|C'|\le2\), and let \(\hat X\subseteq Y_{\mathrm{in}}\cap C\) with \(r\in \hat X\). Then \(\mathrm{Mod}\) has at most \(4^{|C'|}3^{|C\setminus \hat X|}2^{|\hat X|-1}\) reasonable colorings, and they can be listed in time \(4^{|C'|}3^{|C\setminus \hat X|}2^{|\hat X|-1}\tilde n^{O(1)}\) and polynomial space. In particular, if \(|\mathrm{Mod}\setminus \hat X|\le k'+2\), there are at most \(8\cdot3^{k'}2^{|\hat X|}\) of them.

Proof. Fix a spanning tree of \(\tilde G[C]\) rooted at \(r\), and order \(C\) so that every vertex comes after its parent. Let \(\varsigma\) be reasonable. Then \(\varsigma(r)=\mathsf y_1\). Let \(v\in C\setminus\{r\}\) have parent \(p\). If \(\varsigma(p)=\mathsf y_j\), then \(\varsigma(v)\in\{\mathsf y_j,\mathsf z_1,\mathsf z_2\}\), because \(\{\varsigma(p),\varsigma(v)\}\neq\{\mathsf y_1,\mathsf y_2\}\); if \(\varsigma(p)=\mathsf z_j\), then \(\varsigma(v)\in\{\mathsf z_j,\mathsf y_1,\mathsf y_2\}\). If moreover \(v\in \hat X\subseteq Y_{\mathrm{in}}\), then \(\varsigma(v)\in\{\mathsf y_1,\mathsf y_2\}\), which leaves one option when \(\varsigma(p)\) is a \(\mathsf y\)-state and two when it is a \(\mathsf z\)-state. So, given the values on the earlier vertices, \(\varsigma(v)\) lies in an explicit set of at most \(3\) values, and at most \(2\) if \(v\in \hat X\). The vertices of \(C'\) take at most \(4\) values. Enumerating all choices in this order lists a superset of the reasonable colorings, of size at most \(4^{|C'|}3^{|C\setminus \hat X|}2^{|\hat X|-1}\); testing each candidate takes polynomial time, and only the current candidate is stored.

For the last claim, put \(n_1=|C\setminus \hat X|\) and \(n_2=|C'|\). Then \(n_1+n_2=|\mathrm{Mod}\setminus \hat X|\le k'+2\) and \(n_2\le2\), so \(4^{n_2}3^{n_1}=(4/3)^{n_2}3^{n_1+n_2}\le(16/9)3^{k'+2}=16\cdot3^{k'}\), and \(16\cdot3^{k'}2^{|\hat X|-1}=8\cdot3^{k'}2^{|\hat X|}\).  ∎

A structured modulator for \((\tilde G,\hat X,k')\), where \(\hat X\subseteq V(\tilde G)\) is nonempty and \(k'\ge0\), is a set \(\mathrm{Mod}=C\cup C'\) such that \(\tilde G-\mathrm{Mod}\) is a forest, \(\tilde G[C]\) is connected, \(\hat X\subseteq C\), \(|C'|\le2\), \(C\cap C'=\emptyset\) and \(|\mathrm{Mod}\setminus \hat X|\le k'+2\).

Lemma D.9.6 (The test).  Let \(\tilde G\) be a graph with \(\tilde n\) vertices, let \(\emptyset\ne\hat X\subseteq Y_{\mathrm{in}}\subseteq V(\tilde G)\), let \(k'\ge0\), and let \(\mathrm{Mod}\) be a structured modulator for \((\tilde G,\hat X,k')\). One can decide in time \(3^{k'}2^{|\hat X|}\tilde n^{O(1)}\) and polynomial space, with one-sided error, whether \(\tilde G\) has a connected feedback vertex set \(Y\supseteq Y_{\mathrm{in}}\) with \(|Y|\le|\hat X|+k'\): the algorithm never accepts if there is no such set, and it accepts with probability at least \(1/2\) if there is one.

Proof. Fix \(r\in\hat X\). For every \(\eta\) with \(|Y_{\mathrm{in}}|\le\eta\le\min(|\hat X|+k',\tilde n)\), run the algorithm of (CC1) on \((\tilde G,Y_{\mathrm{in}},\eta)\) with the root \(r\) and the tree decomposition of (CC3), computing the count as the sum of the core evaluations over the reasonable colorings of \(\mathrm{Mod}\), listed by Lemma D.9.5; accept if one of these runs accepts. By the remark before Lemma D.9.5, each run decides as the algorithm of (CC1) does. Hence no run accepts if there is no set \(Y\) as in the lemma, and if there is one, the run with \(\eta=|Y|\), which lies in the range tried, accepts with probability at least \(1/2\). There are at most \(\tilde n+1\) runs, each sums at most \(8\cdot3^{k'}2^{|\hat X|}\) core evaluations by Lemma D.9.5, and each core evaluation takes polynomial time and space by (CC3); the sum modulo \(2\) is accumulated coloring by coloring.  ∎

D.9.4 The extension algorithm

Restatement of Lemma 11.7.

The extension algorithm contracts the sample and then follows the iterative compression of the proof of Theorem B.3 of (Cygan et al. 2011) along a contraction sequence, keeping track of the images of the forced vertices. Lemma D.9.4 guarantees that every intermediate graph has a solution of no larger excess, and the previous solution, expanded, is a structured modulator for Lemma D.9.6. Running times are stated for every outcome of the coins, as Corollary 3.15 requires. The extension algorithm solves a forced problem, in which the sample has been contracted: given a connected graph \(\tilde G\) with \(\tilde n\) vertices, a nonempty set \(\hat X\subseteq V(\tilde G)\), weights \(\tilde w\colon V(\tilde G)\to\{0,\dots,w_{\max}\}\) with \(\tilde w=0\) on \(\hat X\), and integers \(k'\ge0\) (the number of further vertices) and \(\theta'\), find a connected feedback vertex set \(A\supseteq\hat X\) of \(\tilde G\) with \(|A\setminus\hat X|\le k'\) and \(\tilde w(A)\le\theta'\); we call such an \(A\) a solution of the forced instance \((\tilde G,\hat X,\tilde w,k',\theta')\). The weights are carried so that Appendix D.10 can reuse the following wrapper, which turns a procedure for the forced problem into an extension algorithm. For \(X=\emptyset\) it guesses one solution vertex, as in Remark A.9 of (Cygan et al. 2011).

Lemma D.9.7 (From the forced problem to the extension problem).  Let \(G\) be a connected graph with \(n\) vertices, \(w\colon V(G)\to\{0,\dots,w_{\max}\}\), \(\theta\) an integer, \(X\subseteq V(G)\) and \(k\ge|X|\), and put \(k'=k-|X|\). Let \(p\in(0,1]\), and let \(\mathrm{Forced}\) be a randomized procedure that, given a forced instance \((\tilde G,\hat X,\tilde w,k'',\theta')\) with \(\tilde n\le n\), runs in polynomial space and, for every outcome of its coins, in time \(3^{k''}2^{|\hat X|}(\tilde n+w_{\max})^{O(1)}\), outputs only solutions of the instance, and outputs one with probability at least \(p\) whenever one exists. Consider the following algorithm.

1.  If \(X\neq\emptyset\): compute the contracted graph \(G_X\) with forced set \(\hat X\), put \(\tilde w=w\) on \(V(G)\setminus X\) and \(\tilde w=0\) on \(\hat X\), and run \(\mathrm{Forced}\) on \((G_X,\hat X,\tilde w,k',\theta-w(X))\). If it outputs \(A\), accept with \(Y=A\setminus\hat X\); otherwise reject.

2.  If \(X=\emptyset\): if \(G\) is a forest, accept with \(Y=\emptyset\) when \(\theta\ge0\) and reject otherwise. If \(G\) is not a forest and \(k'=0\), reject. Otherwise, for every \(v\in V(G)\), let \(\tilde w_v\) agree with \(w\) except that \(\tilde w_v(v)=0\), and run \(\mathrm{Forced}\) on \((G,\{v\},\tilde w_v,k'-1,\theta-w(v))\); if one run outputs \(A\), accept with \(Y=A\). Otherwise reject.

The algorithm runs in polynomial space and, for every outcome of its coins, in time \(3^{k'}2^{\operatorname{comp}(G[X])}(n+w_{\max})^{O(1)}\). If it accepts, then \(Y\subseteq V(G)\setminus X\), \(|Y|\le k'\), and \(X\cup Y\) is a connected feedback vertex set of \(G\) with \(w(X\cup Y)\le\theta\); if such a set \(Y\) exists, it accepts with probability at least \(p\).

Proof. Case \(X\neq\emptyset\). The graph \(G_X\) is connected with at most \(n\) vertices, and \(\tilde w=0\) on \(\hat X\). For \(Y\subseteq V(G)\setminus X\), by Lemma D.9.3(2) the set \(X\cup Y\) is a connected feedback vertex set of \(G\) if and only if \(\hat X\cup Y\) is one of \(G_X\), and \(w(X\cup Y)=w(X)+\tilde w(\hat X\cup Y)\). So the sets \(Y\) in question are exactly the sets \(A\setminus\hat X\) for the sets \(A\) that \(\mathrm{Forced}\) may output on \((G_X,\hat X,\tilde w,k',\theta-w(X))\), and the claims follow from the properties of \(\mathrm{Forced}\), with \(|\hat X|=\operatorname{comp}(G[X])\).

Case \(X=\emptyset\). If \(G\) is a forest, then \(\emptyset\) is a connected feedback vertex set, and no set has smaller weight than \(w(\emptyset)=0\), so the answer is correct. Otherwise every connected feedback vertex set is nonempty. If a run for \(v\) outputs \(A\), then \(A\ni v\) is a connected feedback vertex set with \(|A|\le1+(k'-1)\) and \(w(A)=\tilde w_v(A)+w(v)\le\theta\). Conversely, let \(Y^*\) be a connected feedback vertex set with \(|Y^*|\le k'\) and \(w(Y^*)\le\theta\). Then \(Y^*\neq\emptyset\), so \(k'\ge1\); for \(v\in Y^*\), the set \(Y^*\) contains \(\{v\}\), has \(|Y^*\setminus\{v\}|\le k'-1\) and \(\tilde w_v(Y^*)=w(Y^*)-w(v)\le\theta-w(v)\), so the run for \(v\) outputs a set with probability at least \(p\). The time is \(n\cdot3^{k'-1}2^1(n+w_{\max})^{O(1)}\le3^{k'}(n+w_{\max})^{O(1)}\), with \(\operatorname{comp}(G[\emptyset])=0\).  ∎

Proof of Lemma 11.7. The forced problem. We first solve the forced problem without weights: given a connected graph \(\tilde G\) with \(\tilde n\le n\) vertices, a nonempty set \(\hat X\subseteq V(\tilde G)\) and \(k'\ge0\), find a connected feedback vertex set \(A\supseteq\hat X\) of \(\tilde G\) with \(|A\setminus\hat X|\le k'\). Compute graphs \(\tilde G_{\tilde n}=\tilde G,\tilde G_{\tilde n-1},\dots,\tilde G_1\) with \(\tilde G_i=\tilde G_{i+1}/u_iv_i\) for an edge \(u_iv_i\) of \(\tilde G_{i+1}\) and new vertex \(w_i\); all of them are connected by Lemma D.9.3(1). Let \(\hat X_i\) be the image of \(\hat X\) in \(\tilde G_i\); then \(|\hat X_i|\le|\hat X|\), \(\tilde G_1\) has one vertex, \(\hat X_1=V(\tilde G_1)\), and we set \(A_1=V(\tilde G_1)\). For \(i=1,\dots,\tilde n-1\), given a connected feedback vertex set \(A_i\supseteq\hat X_i\) of \(\tilde G_i\) with \(|A_i\setminus\hat X_i|\le k'\), let \(\mathrm{Mod}=(A_i\setminus\{w_i\})\cup\{u_i,v_i\}\) in \(\tilde G_{i+1}\), and let \(C=\mathrm{Mod}\), \(C'=\emptyset\) if \(w_i\in A_i\), and \(C=A_i\), \(C'=\{u_i,v_i\}\) otherwise. Start with \(Y_{\mathrm{in}}=\hat X_{i+1}\) and go through the vertices \(v\notin\hat X_{i+1}\) in a fixed order: run the test of Lemma D.9.6 on \((\tilde G_{i+1},Y_{\mathrm{in}}\cup\{v\},\hat X_{i+1},k',\mathrm{Mod})\) independently \(\lceil\log(4\tilde n^2)\rceil\) times, and if one run accepts, add \(v\) to \(Y_{\mathrm{in}}\). Afterward, if \(Y_{\mathrm{in}}\) is a connected feedback vertex set of \(\tilde G_{i+1}\) with \(|Y_{\mathrm{in}}\setminus\hat X_{i+1}|\le k'\), set \(A_{i+1}=Y_{\mathrm{in}}\); otherwise stop without a solution. Output \(A_{\tilde n}\). This is the self-reduction of the proof of (Cygan et al. 2022 Theorem 5.1) and of Theorem B.3 of (Cygan et al. 2011), with the forced vertices kept in \(Y_{\mathrm{in}}\) throughout.

The modulator. Write \(u,v,w\) for \(u_i,v_i,w_i\). The graphs \(\tilde G_{i+1}-\{u,v\}\) and \(\tilde G_i-w\) are identical, so \(\tilde G_{i+1}-\mathrm{Mod}=\tilde G_i-(A_i\cup\{w\})\) is an induced subgraph of the forest \(\tilde G_i-A_i\). Every forced vertex of \(\tilde G_{i+1}\) other than \(u,v\) is a forced vertex of \(\tilde G_i\) other than \(w\), hence lies in \(A_i\setminus\{w\}\). If \(w\in A_i\), then \(\tilde G_{i+1}[\mathrm{Mod}]\) is connected by Lemma D.9.3(1), because \(\tilde G_{i+1}[\mathrm{Mod}]/uv=\tilde G_i[A_i]\), and \(\hat X_{i+1}\subseteq\mathrm{Mod}=C\). If \(w\notin A_i\), then \(w\notin\hat X_i\), so \(u,v\notin\hat X_{i+1}\), \(\hat X_{i+1}\subseteq A_i=C\), and \(\tilde G_{i+1}[A_i]=\tilde G_i[A_i]\) is connected. Finally \(|\mathrm{Mod}\setminus\hat X_{i+1}|\le|A_i\setminus\hat X_i|+2\le k'+2\). So \(\mathrm{Mod}\) is a structured modulator for \((\tilde G_{i+1},\hat X_{i+1},k')\), and \(\hat X_{i+1}\subseteq Y_{\mathrm{in}}\cup\{v\}\) in every test.

Correctness. Every set \(A_i\) is checked before it is kept, so an output is a connected feedback vertex set \(A\supseteq\hat X\) of \(\tilde G\) with \(|A\setminus\hat X|\le k'\). Suppose such a set exists. By Lemma D.9.4, applied along the contraction sequence, so does every \(\tilde G_i\) for \(\hat X_i\). Step \(i\) is the self-reduction of Lemma D.10.1 with \(\Omega=V(\tilde G_{i+1})\), \(Z_0=\hat X_{i+1}\), the family \(\mathcal P\) of the connected feedback vertex sets \(Y\supseteq\hat X_{i+1}\) of \(\tilde G_{i+1}\) with \(|Y\setminus\hat X_{i+1}|\le k'\), which is nonempty, the test \(\mathsf T(Y_{\mathrm{in}})\) of Lemma D.9.6 on \((\tilde G_{i+1},Y_{\mathrm{in}},\hat X_{i+1},k',\mathrm{Mod})\), \(\delta=1/(4\tilde n)\) and \(\rho=\lceil\log(4\tilde n^2)\rceil\ge\log(|\Omega|/\delta)\), since \(|\Omega|\le\tilde n\). By Lemma D.9.6, the test rejects \(Y_{\mathrm{in}}\) if no member of \(\mathcal P\) contains it (for \(Y\supseteq\hat X_{i+1}\) the conditions \(|Y|\le|\hat X_{i+1}|+k'\) and \(|Y\setminus\hat X_{i+1}|\le k'\) agree), and with probability at least \(1/2\) if one does. So step \(i\) fails to produce a member of \(\mathcal P\) with probability at most \(1/(4\tilde n)\); there are fewer than \(\tilde n\) steps, so by the union bound a solution is found with probability at least \(3/4\). Each step runs at most \(\tilde n\lceil\log(4\tilde n^2)\rceil\) tests, each in time \(3^{k'}2^{|\hat X_{i+1}|}\tilde n^{O(1)}\le3^{k'}2^{|\hat X|}\tilde n^{O(1)}\) and polynomial space by Lemma D.9.6; only the current \(A_i\) is stored between steps. So the forced problem is solved in time \(3^{k'}2^{|\hat X|}\tilde n^{O(1)}\) and polynomial space.

The algorithm \(\mathcal{A}\). Let \(\mathcal{A}\) be the algorithm of Lemma D.9.7 with \(w\equiv0\), \(\theta=0\) and, as \(\mathrm{Forced}\), the procedure above, which ignores \(\tilde w\) and \(\theta'\); every call then has \(\tilde w\equiv0\) and \(\theta'=0\), so every output satisfies \(\tilde w(A)\le\theta'\), and the hypotheses of Lemma D.9.7 hold with \(w_{\max}=0\) and \(p=3/4\). The lemma gives the time, space and correctness claims. The running-time exponent (22) is nonnegative and computable in polynomial time; comparisons of \(g(G,X,k)\) with numbers of the form \(i\log3+j\) with integers \(i,j\), which are the only ones the algorithms below make, reduce to integer comparisons.  ∎

D.9.5 Tightness of the base \(1.8\)

This subsection proves Propositions 11.9 and 11.10. The proof of Proposition 11.9 uses the following evaluation of the infimum over \(y\) of the function \(\phi_{3,\sigma}\) of (21).

Lemma D.9.8 (The critical fraction for \(c=3\)).  For \(\sigma^*=26/45\), \(\inf_{y\ge0}\phi_{3,\sigma^*}(y)=\log1.8\), attained at \(y=4/5\).

Lean, partial: cc_c3 (formalisation)

Proof. The map \(y\mapsto u=\lambda_{\max}(M_{3y/2})\) is an increasing bijection from \([0,\infty)\) onto \([1,\infty)\). Its inverse is \(\zeta=3y/2=(u^2-u)/(2u-1)\), because \(u\) is a root of \(x^2-(1+2\zeta)x+\zeta\) (and the larger one, as the other root \(\zeta/u=(u-1)/(2u-1)\) is less than \(1\)). Substituting, \(1+y=(2u^2+4u-3)/(3(2u-1))\), and \(\phi_{3,\sigma^*}(y)\ln2=\hat\phi(u)\) with \[\hat\phi(u)=\ln(2u^2+4u-3)-\ln(2u-1)-\ln3-\sigma^*\ln\frac u3\qquad(u\ge1).\] Differentiating and bringing the three terms to a common denominator gives \[u\,\hat\phi'(u)=\frac{4u^2+4u}{2u^2+4u-3}-\frac{2u}{2u-1}-\frac{26}{45} =\frac{(u-3)(76u^2-108u+26)}{45\,(2u^2+4u-3)(2u-1)} .\] The denominator is positive for \(u\ge1\). The quadratic \(76u^2-108u+26\) is negative at \(u=1\) and has exactly one root \(u_1=(54+\sqrt{940})/76\in(1,3)\) in \([1,\infty)\). So \(\hat\phi\) increases on \([1,u_1]\), decreases on \([u_1,3]\) and increases on \([3,\infty)\), and its minimum on \([1,\infty)\) is \(\min(\hat\phi(1),\hat\phi(3))\). Now \(\hat\phi(3)=\ln(27/15)=\ln1.8\), which is the value at \(y=4/5\), and \(\hat\phi(1)=\sigma^*\ln3\). Finally \(\sigma^*\ln3>\ln1.8\) is equivalent to \(5^{45}>3^{64}\), which holds: \(5^{45}=125^{15}>(\tfrac32\cdot81)^{15}=(\tfrac32)^{15}3^{60}\), and \((\tfrac32)^{15}>(\tfrac32)^{12}=(\tfrac{81}{16})^3>5^3>3^4\).  ∎

Restatement of Proposition 11.9.

Proof. (a) By the assumed lower bound on \(g\), \(W_S(y)\le2^{\gamma'}3^{-|S|}\mathcal Z_F(3y/2)\) for \(y\ge0\), by the computation in the proof of Corollary 11.5(a) with the inequality reversed. By Lemma 11.3 with \(\zeta=3y/2\), \(3^{-|S|}\mathcal Z_F(\zeta)\le2(\lambda_{\max}(M_\zeta)/3)^{|S|}\), so, with \(\sigma^*=26/45\), \(W_S(y)/(1+y)^n\le2^{\gamma'+1}2^{-n\phi_{3,\sigma^*}(y)}\le2^{\gamma'+1}(5/9)^n\) by Lemma D.9.8. By Lemma 3.9, \(\max_t\mu_t(S)/\binom nt\le(n+1)\sup_{y\ge0}W_S(y)/(1+y)^n\), which proves (a).

(b) Let \(|S|\le k\le n\). By Proposition 3.10 for \(S\) and \(k\), the expected number of trials of the run on \((I,k)\) that run \(\mathcal{A}\) on a subset of \(S\) is at most \(P_{\mathrm{nec}}\cdot2^{\psi n}\max_{t\le|S|}\mu^{(k)}_t(S)/\binom nt\). As \(g\) is nondecreasing in its budget, (4) gives \(\mu^{(k)}_t(S)\le\mu_t(S)\), and (a) bounds the expectation by \(P_{\mathrm{nec}}\cdot2^{\gamma'+1}(n+1)(2^\psi\cdot5/9)^n\). By Markov’s inequality, this also bounds the probability that at least one such trial occurs.

(c) A run answers yes, or returns a set, only after a trial with some sample \(X\) of size \(t\) in which \(\mathcal{A}\) accepts \((I,X,k)\). Since \(\mathcal{A}\) never accepts a no-instance, some \(Y\subseteq U_I\setminus X\) with \(|Y|\le k-t\) satisfies \(X\cup Y\in\mathcal{F}_I\), and a returned set is such a set \(X\cup Y\). If \(k\le|S|\), then \(|X\cup Y|\le k\le|S|\); if the run returns a set of size at most \(|S|\), then this set is an \(X\cup Y\) of size at most \(|S|\). In both cases \(X\cup Y\) is a set in \(\mathcal{F}_I\) of size at most \(|S|\), so \(X\cup Y=S\) and \(X\subseteq S\).

(d) A run with \(k<|S|\) never answers yes, by item (ii) of Theorem 3.5, since \(S\) is the only set in \(\mathcal{F}_I\) of size at most \(|S|\) and \(|S|>k\). By (c), a yes at \(k=|S|\) or a returned set of size at most \(|S|\) requires a run that runs \(\mathcal{A}\) on a subset of \(S\), and this run has \(k\ge|S|\), because a returned set \(X\cup Y\) has size at most \(|X|+(k-|X|)=k\) and equals \(S\) by the proof of (c). By (b) and the union bound over the at most \(n\) runs with \(k\ge|S|\), the probability is at most \(nP_{\mathrm{nec}}2^{\gamma'+1}(n+1)(2^\psi\cdot5/9)^n\), and \(2^\psi\cdot5/9\le2^{-\varepsilon}\cdot1.8\cdot5/9=2^{-\varepsilon}\).  ∎

Restatement of Proposition 11.10.

Proof. Let \(L\ge2\) and let \(T\subseteq\{0,\dots,L-1\}\) contain \(0\) and \(L-1\). Let \(G_{L,T}\) consist of a path \(v_0v_1\dots v_L\) and, for each \(i\in T\), a vertex \(w_i\) adjacent to \(v_i\) and \(v_{i+1}\), and put \(S=\{v_1,\dots,v_{L-1}\}\). We show that \(S\) is the unique minimum connected feedback vertex set of \(G_{L,T}\), that \(G_{L,T}[S]\) is a path, and that \(g(G_{L,T},X,|S|)=(|S|-|X|)\log3+|X|-e(G_{L,T}[X])\) for every \(X\subseteq S\). The family consists of the graphs with \(L=26j+1\) and \(|T|=19j-2\), \(j\ge1\).

Write \(G=G_{L,T}\). The set \(S\) is a connected feedback vertex set: \(G[S]\) is the path \(v_1\dots v_{L-1}\), and \(G-S\) consists of the edges \(v_0w_0\) and \(v_Lw_{L-1}\) and isolated vertices. Let \(A\) be any connected feedback vertex set. It meets the triangle \(v_0v_1w_0\) in a vertex \(x\) and the triangle \(v_{L-1}v_Lw_{L-1}\) in a vertex \(z\). For \(1\le h\le L-1\), deleting \(v_h\) separates \(\{v_0,\dots,v_{h-1}\}\cup\{w_i:i<h\}\) from \(\{v_{h+1},\dots,v_L\}\cup\{w_i:i\ge h\}\). Either \(v_h\in\{x,z\}\), or \(x\) lies in the first part and \(z\) in the second; in the latter case \(v_h\in A\) because \(G[A]\) is connected. Hence \(S\subseteq A\), so \(S\) is the unique minimum solution. For \(X\subseteq S\), \(G[X]\) is a subgraph of a path, so \(\operatorname{comp}(G[X])=|X|-e(G[X])\), and the formula for \(g\) is (22). With \(L=26j+1\) and \(|T|=19j-2\) (possible, since \(2\le19j-2\le L\)) we get \(|S|=26j\) and \(n=L+1+|T|=45j\), so \(|S|/n=26/45\).

We check the hypotheses of Proposition 11.9 for \(I=G\), the path \(F=G[S]\) and \(\gamma'=0\). The running-time exponent \(g(G,X,k)=(k-|X|)\log3+\operatorname{comp}(G[X])\) is nondecreasing in \(k\), and the formula above gives the lower bound on \(g(G,X,|S|)\) with equality. By Lemma 11.7, the extension algorithm never accepts a no-instance and returns sets \(Y\) such that \(X\cup Y\) is a connected feedback vertex set. Every connected feedback vertex set contains \(S\), so \(S\) is the only one of size at most \(|S|\). The algorithm of Theorem 11.8 outputs \(S\) only if some run returns a set of size at most \(|S|\) (its default output \(V(G)\) has size \(n>|S|\)), so by Proposition 11.9(d) with \(\gamma'=0\) this has probability at most \(2nP_{\mathrm{nec}}(n+1)2^{-\varepsilon n}\), which tends to \(0\) as \(j\to\infty\), because the input length \(N\) is polynomial in \(n\) for the graphs \(G_{L,T}\).  ∎

D.10 Weighted finishing steps

This subsection proves Theorems 11.16 and 11.17. In both cases the sampling analysis is that of Section 11.5, and only the extension algorithm is new. Both extension algorithms turn a randomized decision procedure into one that outputs a witness, by the following self-reduction, which is also the one used in the proof of Lemma 11.7.

Lemma D.10.1 (Witness by self-reduction).  Let \(\Omega\) be a set with \(m\ge1\) elements, let \(\mathcal P\) be a nonempty family of subsets of \(\Omega\), let \(Z_0\) be a subset of some member of \(\mathcal P\), let \(0<\delta<1\) and let \(\rho\ge\log(m/\delta)\) be an integer. Let \(\mathsf T\) be a randomized test that, on input \(Z\subseteq\Omega\), never accepts if no member of \(\mathcal P\) contains \(Z\), and accepts with probability at least \(1/2\) otherwise, with independent coins in every run. Starting from \(Z=Z_0\), go through the elements \(v\in\Omega\setminus Z_0\) in a fixed order and add \(v\) to \(Z\) if one of \(\rho\) independent runs of \(\mathsf T\) on \(Z\cup\{v\}\) accepts. Then \(\mathsf T\) is run at most \(m\rho\) times, and the final set \(Z\) is a member of \(\mathcal P\) with probability at least \(1-\delta\).

Proof. Throughout, \(Z\) is contained in a member of \(\mathcal P\): this holds for \(Z_0\), and \(v\) is added only after an accepting run on \(Z\cup\{v\}\), which by the soundness of \(\mathsf T\) is contained in a member. For \(v\in\Omega\setminus Z_0\) let \(E_v\) be the event that, when \(v\) is processed with the current set \(Z\), the set \(Z\cup\{v\}\) is contained in a member of \(\mathcal P\) and all \(\rho\) runs reject. Conditioned on the history before \(v\) is processed, each run then rejects with probability at most \(1/2\), independently, so \(\Pr[E_v]\le2^{-\rho}\le\delta/m\), and by the union bound some \(E_v\) occurs with probability at most \(\delta\). Suppose that no \(E_v\) occurs, and let \(P\in\mathcal P\) contain the final set \(Z\). If some \(v\in P\setminus Z\) existed, then \(v\notin Z_0\), and when \(v\) was processed the current set was contained in \(Z\), so the tested set was contained in \(P\) and all runs rejected it, that is, \(E_v\) occurred. Hence \(Z=P\). There are at most \(m\) elements to process, each with \(\rho\) runs.  ∎

D.10.1 Maximum-Weight Connected \(k\)-Subgraph

Restatement of Theorem 11.16.

The finishing step is the algebraic sieve of Björklund, Kaski and Kowalik (Björklund, Kaski, and Kowalik 2016) for Graph Motif, with one more indeterminate that records the weight, as they do for the edit costs of Closest Graph Motif. Their cost-constrained lemma (Björklund, Kaski, and Kowalik 2016 Lemma 3) lets the coloring vary, whereas we need the coloring fixed and a cost attached to each vertex; so we derive the variant we need from the proofs of their Lemmas 1 and 2. As in the proof of Lemma 11.11, the components of the sample are contracted and given fresh colors, which the motif demands exactly once.

Notation from (Björklund, Kaski, and Kowalik 2016).

Let \(H\) be a graph with vertex set \(\{1,\dots,n_H\}\), let \(D\subseteq V(H)\), let \(w'\colon V(H)\to\{-w_{\max},\dots,w_{\max}\}\) and let \(p\) be an integer with \(\max(1,|D|)\le p\le n_H\). A branching walk in \(H\) is a pair \((T,\varphi)\) of an ordered rooted tree \(T\), whose vertices are numbered in preorder, and a homomorphism \(\varphi\colon V(T)\to V(H)\); it is simple if \(\varphi\) is injective, properly ordered if siblings \(a<b\) satisfy \(\varphi(a)<\varphi(b)\), it has size \(|V(T)|\), and it spans \(\varphi(V(T))\). With a variable \(x_u\) for every vertex \(u\) and variables \(y_{(u,u')},y_{(u',u)}\) for every edge \(uu'\), the fingerprint of a walk is \(\prod_{ab\in E(T),\,a<b}y_{(\varphi(a),\varphi(b))}x_{\varphi(b)}\), and \[P_p(\vec x,\vec y)=\sum_{s\in V(H)}x_s\sum_{(T,\varphi)}\prod_{ab\in E(T),\,a<b}y_{(\varphi(a),\varphi(b))}x_{\varphi(b)},\] the inner sum over the properly ordered branching walks of size \(p\) that start at \(s\) ((Björklund, Kaski, and Kowalik 2016, sec. 3.2)). Every monomial of \(P_p\) has degree \(p\) in \(\vec x\) and degree \(p-1\) in \(\vec y\). By (Björklund, Kaski, and Kowalik 2016 Lemma 7), a monomial of \(P_p\) is multilinear in \(\vec x\) if and only if it comes from a simple walk, and a simple walk can be reconstructed from its monomial; so the monomials of \(P_p\) that are multilinear in \(\vec x\) are in bijection with the simple properly ordered branching walks of size \(p\), each has coefficient \(1\), and its \(\vec x\)-part is \(\prod_{u\in K}x_u\) for the set \(K\) that the walk spans, which induces a connected subgraph. Conversely, every set \(K\) with \(|K|=p\) and \(H[K]\) connected is spanned by such a walk: take a spanning tree of \(H[K]\), root it, order siblings by their vertex numbers and number the vertices in preorder.

Color the vertices of \(H\) as follows: each \(u\in D\) gets its own color \(c_u\) with multiplicity \(m(c_u)=1\), and every other vertex gets the color \(\star\) with multiplicity \(m(\star)=p-|D|\). Give each color \(c\) a set \(\Sigma_c\) of \(m(c)\) shades, these sets pairwise disjoint, put \(\Sigma=\bigcup_c\Sigma_c\), so \(|\Sigma|=p\), and let \(\Lambda\) be a set of \(p\) labels. A set \(K\subseteq V(H)\) is properly colored if every color \(c\) occurs on at most \(m(c)\) vertices of \(K\). A set \(K\) with \(|K|=p\) is properly colored if and only if \(D\subseteq K\): if \(D\subseteq K\), each \(c_u\) occurs once and \(\star\) occurs \(p-|D|\) times; conversely, at most \(p-|D|\) vertices of \(K\) have color \(\star\), so at least \(|D|\) vertices of \(K\) lie in \(D\), that is, \(D\subseteq K\). The cost of a vertex is \(\operatorname{cost}(u)=w_{\max}-w'(u)\in\{0,\dots,2w_{\max}\}\), and \(\operatorname{cost}(K)=\sum_{u\in K}\operatorname{cost}(u)=p\,w_{\max}-w'(K)\) for \(|K|=p\).

Work over the field \(\mathbb F=\mathbb F_{2^b}\). Introduce variables \(v_{u,d}\) for \(u\in V(H)\) and \(d\in\Sigma_{\mathrm{col}(u)}\), variables \(\zeta_{d,j}\) for \(d\in\Sigma\) and \(j\in\Lambda\), and one more variable \(\lambda\). Put \(\upsilon_{u,j}=\sum_{d\in\Sigma_{\mathrm{col}(u)}}v_{u,d}\zeta_{d,j}\), \(\upsilon^A_u=\sum_{j\in A}\upsilon_{u,j}\) for \(A\subseteq\Lambda\), and \[\tag{35} Q(\vec v,\vec\zeta,\vec y,\lambda)=\sum_{A\subseteq\Lambda}P_p\bigl((\lambda^{\operatorname{cost}(u)}\upsilon^A_u)_{u\in V(H)},\vec y\bigr).\] This is the sieve (2) of (Björklund, Kaski, and Kowalik 2016) applied to \(P_p(\lambda^{\operatorname{cost}(1)}x_1,\dots, \lambda^{\operatorname{cost}(n_H)}x_{n_H},\vec y)\). Write \(Q=\sum_\sigma Q_\sigma(\vec v,\vec\zeta,\vec y)\lambda^\sigma\).

Lemma D.10.2 (Weighted sieve identity).  Let \(H\), \(D\), \(w'\), \(p\) and the coloring be as above, and let \(\sigma\ge0\) be an integer. The polynomial \(Q_\sigma\) is nonzero if and only if some set \(K\subseteq V(H)\) with \(|K|=p\), \(D\subseteq K\) and \(H[K]\) connected has \(\operatorname{cost}(K)=\sigma\), that is, \(w'(K)=pw_{\max}-\sigma\). Moreover \(Q_\sigma\) has total degree at most \(3p-1\), and \(Q_\sigma=0\) for \(\sigma>2pw_{\max}\).

Proof. The idea is that the sieve of (Björklund, Kaski, and Kowalik 2016) acts on each monomial of \(P_p\) separately, and the substitution \(x_u\mapsto\lambda^{\operatorname{cost}(u)}x_u\) only multiplies each monomial by a power of \(\lambda\) determined by its \(\vec x\)-part. Write \(P_p=\sum_m a_m\vec x^{\,\delta_m}\vec y^{\,\epsilon_m}\) with distinct exponent pairs \((\delta_m,\epsilon_m)\) and nonzero coefficients \(a_m\). Then \[Q=\sum_m a_m\,\lambda^{\sum_u\operatorname{cost}(u)\delta_{m,u}}\,\vec y^{\,\epsilon_m}\sum_{A\subseteq\Lambda}\prod_{u}\bigl(\upsilon^A_u\bigr)^{\delta_{m,u}} .\] Fix \(m\) and write \(\delta=\delta_m\); then \(\sum_u\delta_u=p=|\Lambda|\). The proofs of Lemmas 1 and 2 of (Björklund, Kaski, and Kowalik 2016) (equations (5)–(8)) show, in characteristic \(2\), that \(\sum_A\prod_u(\upsilon^A_u)^{\delta_u}=0\) unless \(\delta\in\{0,1\}^{V(H)}\); that argument uses only \(\upsilon^A_u=\sum_{j\in A}\upsilon_{u,j}\) and \(\sum_u\delta_u=|\Lambda|\). If \(\delta\) is the indicator vector of a set \(K\), then by (8) and (9) of (Björklund, Kaski, and Kowalik 2016) and the paragraph after (9), \[\sum_{A\subseteq\Lambda}\prod_{u\in K}\upsilon^A_u=R_K:=\sum_{h}\ \sum_{g}\ \prod_{u\in K}v_{u,h(u)}\zeta_{h(u),g(u)},\] where \(h\) ranges over the injective maps \(K\to\Sigma\) with \(h(u)\in\Sigma_{\mathrm{col}(u)}\) and \(g\) over the bijections \(K\to\Lambda\); the triple \((K,h,g)\) can be read off from the monomial \(\prod_{u\in K}v_{u,h(u)}\zeta_{h(u),g(u)}\), so these monomials are distinct and \(R_K\neq0\) if and only if such an \(h\) exists, that is, if and only if \(K\) is properly colored.

By the facts quoted from (Björklund, Kaski, and Kowalik 2016 Lemma 7), the monomials of \(P_p\) that are multilinear in \(\vec x\) are \(\prod_{u\in K_W}x_u\,\vec y^{\,\epsilon_W}\) with coefficient \(1\), one for each simple properly ordered branching walk \(W\) of size \(p\), where \(K_W\) is the set that \(W\) spans and \(\vec y^{\,\epsilon_W}\) the \(\vec y\)-part of its fingerprint. Hence \[Q_\sigma=\sum_{W\,:\,\operatorname{cost}(K_W)=\sigma,\ K_W\text{ properly colored}}\vec y^{\,\epsilon_W}\,R_{K_W}.\] The monomials \(\vec y^{\,\epsilon_W}\prod_{u\in K_W}v_{u,h(u)}\zeta_{h(u),g(u)}\) over all such \(W\) and all \((h,g)\) are pairwise distinct: the \(\vec v\)-variables determine \(K_W\) and \(h\), the \(\vec\zeta\)-variables then determine \(g\), and \(K_W\) together with \(\vec y^{\,\epsilon_W}\) is the monomial of \(W\) in \(P_p\), which determines \(W\). So no cancellation occurs, and \(Q_\sigma\neq0\) if and only if some simple walk of size \(p\) spans a properly colored set \(K\) with \(\operatorname{cost}(K)=\sigma\). By the facts quoted above and the characterization of properly colored sets, this holds if and only if some \(K\) with \(|K|=p\), \(D\subseteq K\) and \(H[K]\) connected has \(\operatorname{cost}(K)=\sigma\). Each monomial of \(Q_\sigma\) has degree \(p-1\) in \(\vec y\) and \(2p\) in \((\vec v,\vec\zeta)\), and \(\operatorname{cost}(K)\le2pw_{\max}\).  ∎

Algorithm D.10.3 (Weighted sieve).  \(H\), \(D\), \(w'\), \(p\) as above and an integer \(\theta'\). If \(p>n_H\) or \(\theta'>pw_{\max}\), reject. Let \(b=\lceil\log(6pw_{\max})\rceil\), and find an irreducible polynomial of degree \(b\) over \(\mathbb F_2\) by trying all \(2^b\) monic polynomials of degree \(b\) and testing each by division by all polynomials of degree between \(1\) and \(\lfloor b/2\rfloor\); it defines \(\mathbb F=\mathbb F_{2^b}\). Draw values for all variables \(\vec v,\vec\zeta,\vec y\) independently and uniformly from \(\mathbb F\). For each of \(D_\lambda+1\) distinct nonzero elements \(\lambda_0\in\mathbb F\), where \(D_\lambda=2pw_{\max}\), compute \(Q(\vec v,\vec\zeta,\vec y,\lambda_0)\) by evaluating, for every \(A\subseteq\Lambda\), the polynomial \(P_p\) at the point \(x_u=\lambda_0^{\operatorname{cost}(u)}\upsilon^A_u\) with the recurrence (11)–(13) of (Björklund, Kaski, and Kowalik 2016), and summing over \(A\). Recover the coefficients \(Q_\sigma(\vec v,\vec\zeta,\vec y)\), \(0\le\sigma\le D_\lambda\), of the univariate polynomial \(\lambda\mapsto Q(\vec v,\vec\zeta,\vec y,\lambda)\) by Lagrange interpolation. Accept if \(Q_\sigma(\vec v,\vec\zeta,\vec y)\neq0\) for some \(\sigma\le pw_{\max}-\theta'\); otherwise reject.

Lemma D.10.4 (Weighted sieve, guarantees).  Let \(H\) have \(n_H\) vertices. Algorithm D.10.3 runs in polynomial space and, for every outcome of its coins, in time \(2^p(n_Hw_{\max})^{O(1)}\). It rejects if no \(K\subseteq V(H)\) with \(|K|=p\), \(D\subseteq K\) and \(H[K]\) connected has \(w'(K)\ge\theta'\), and if such a \(K\) exists, it accepts with probability at least \(1/2\).

Proof. Since \(w_{\max}\ge1\), we have \(2^b\ge6pw_{\max}\ge\max(6p,D_\lambda+2)\), so the field has \(D_\lambda+1\) nonzero distinct elements. By Lemma D.10.2, \(\lambda\mapsto Q(\vec v,\vec\zeta,\vec y,\lambda)\) is a polynomial of degree at most \(D_\lambda\) whose coefficients are the values \(Q_\sigma(\vec v,\vec\zeta,\vec y)\), so the interpolation recovers them. If the algorithm accepts, some \(Q_\sigma\) with \(\sigma\le pw_{\max}-\theta'\) is a nonzero polynomial, and Lemma D.10.2 gives a set \(K\) as in the lemma with \(w'(K)=pw_{\max}-\sigma\ge\theta'\). Conversely, let \(K^*\) be such a set and \(\sigma^*=\operatorname{cost}(K^*)=pw_{\max}-w'(K^*)\le pw_{\max}-\theta'\). By Lemma D.10.2, \(Q_{\sigma^*}\) is a nonzero polynomial of total degree at most \(3p-1\), so by the lemma of DeMillo, Lipton, Schwartz and Zippel in the form of (Björklund, Kaski, and Kowalik 2016 Lemma 8) its value at a uniformly random point is zero with probability at most \((3p-1)/2^b<1/2\).

The search for the irreducible polynomial takes time \(2^{3b/2}b^{O(1)}=(pw_{\max})^{O(1)}\), as \(2^b<12pw_{\max}\), and every field operation takes time \(b^{O(1)}\). For fixed \(\lambda_0\) and \(A\), the point \((\lambda_0^{\operatorname{cost}(u)}\upsilon^A_u)_u\) is computed with \(O(p^2n_H+n_Hw_{\max})\) field operations from the values \(\upsilon_{u,j}\), which are computed once, and \(P_p\) is evaluated with \(O(p^2|E(H)|)\) field operations by the recurrence (11)–(13) ((Björklund, Kaski, and Kowalik 2016, sec. 3.5)). There are \((D_\lambda+1)2^p\) such pairs, and the interpolation takes \(D_\lambda^{O(1)}\) field operations. The algorithm stores the \(D_\lambda+1\) sums and the tables of one evaluation of the recurrence, which is polynomial space.  ∎

The sieve only decides. A set is found by deleting vertices, as follows.

Algorithm D.10.5 (Weighted finishing with a witness).  \(H\), \(D\), \(w'\), \(p\), \(\theta'\) as in Algorithm D.10.3. Put \(\rho=\lceil\log(4n_H)\rceil\) and \(H_{\mathrm{cur}}=H\). For every \(u\in V(H)\setminus D\) in a fixed order, run Algorithm D.10.3 on \((H_{\mathrm{cur}}-u,D,w',p,\theta')\) independently \(\rho\) times, and if one run accepts, replace \(H_{\mathrm{cur}}\) by \(H_{\mathrm{cur}}-u\). Afterward, if \(K=V(H_{\mathrm{cur}})\) satisfies \(|K|=p\), \(H[K]\) connected and \(w'(K)\ge\theta'\), output \(K\); otherwise reject.

Lemma D.10.6 (Weighted finishing step).  Algorithm D.10.5 runs in polynomial space and, for every outcome of its coins, in time \(2^p(n_Hw_{\max})^{O(1)}\). Every output \(K\) satisfies \(|K|=p\), \(D\subseteq K\), \(H[K]\) connected and \(w'(K)\ge\theta'\). If such a set exists, the algorithm outputs one with probability at least \(3/4\).

Proof. The output is checked explicitly. Suppose that a set as in the lemma exists. If \(V(H)=D\), then this set is \(V(H)\), the loop is empty, and the final check outputs it. Otherwise, with \(Z\) the set of deleted vertices, so that \(H_{\mathrm{cur}}=H-Z\), the algorithm is the self-reduction of Lemma D.10.1 with \(\Omega=V(H)\setminus D\), \(Z_0=\emptyset\), the family \(\mathcal P\) of the sets \(V(H)\setminus K\) over the sets \(K\) with \(|K|=p\), \(D\subseteq K\), \(H[K]\) connected and \(w'(K)\ge\theta'\), the test \(\mathsf T(Z)\) given by Algorithm D.10.3 on \((H-Z,D,w',p,\theta')\), \(\delta=1/4\) and \(\rho=\lceil\log(4n_H)\rceil\ge\log(|\Omega|/\delta)\). The family is nonempty by assumption, and \(Z_0\) lies in every member. By Lemma D.10.4, the test rejects \(Z\) if \(H-Z\) contains no such set \(K\), that is, if \(Z\) lies in no member of \(\mathcal P\), and it accepts with probability at least \(1/2\) if this holds, since then \(K\subseteq V(H-Z)\) and \((H-Z)[K]=H[K]\). So, with probability at least \(3/4\), the final \(Z\) is \(V(H)\setminus K\) for such a \(K\), and \(K\) is output. There are at most \(n_H\rho\) runs of Algorithm D.10.3, each within the bounds of Lemma D.10.4.  ∎

The extension problem.

For an instance \(I=(G,w,k)\) and an integer \(\theta\), let \(\Phi_\theta\) be the implicit set system with universe \(V(G)\) whose solutions are the connected \(k\)-sets \(S\) with \(w(S)\ge\theta\). Membership is decided in polynomial time, and every solution has size \(k\). For a nonempty \(X\subseteq V(G)\), let \(X_1,\dots,X_q\) be the vertex sets of the components of \(G[X]\), and let \(G_X\) be obtained from \(G\) by contracting each \(X_i\) to a new vertex \(d_i\), as in Appendix D.9.2; here \(G\) need not be connected, and \(\hat X=\{d_1,\dots,d_q\}\) is an independent set of \(G_X\). For \(X=\emptyset\) put \(G_X=G\) and \(\hat X=\emptyset\). Define \(w_X\colon V(G_X)\to\{-w_{\max},\dots,w_{\max}\}\) by \(w_X=w\) on \(V(G)\setminus X\) and \(w_X(d_i)=0\).

For \(Y\subseteq V(G)\setminus X\), the graph \(G_X[\hat X\cup Y]\) is obtained from \(G[X\cup Y]\) by contracting the connected sets \(X_1,\dots,X_q\): the vertex \(d_i\) is adjacent to \(y\in Y\) exactly when \(y\) has a neighbor in \(X_i\), edges inside \(Y\) are unchanged, and there are no edges between different \(X_i\). So, by Lemma D.9.3(1), \(G[X\cup Y]\) is connected if and only if \(G_X[\hat X\cup Y]\) is. Moreover \(|\hat X\cup Y|=|Y|+q\) and \(w_X(\hat X\cup Y)=w(X\cup Y)-w(X)\). Every set \(K\subseteq V(G_X)\) with \(\hat X\subseteq K\) is \(\hat X\cup Y\) for \(Y=K\setminus\hat X\subseteq V(G)\setminus X\). Consequently, for every integer \(\ell\ge0\), some \(Y\subseteq V(G)\setminus X\) with \(|Y|\le\ell-|X|\) makes \(X\cup Y\) a solution of \(\Phi_\theta\) if and only if \(|X|\le k\le\ell\) and \(G_X\) has a set \(K\supseteq\hat X\) with \(|K|=k-|X|+q\), \(G_X[K]\) connected and \(w_X(K)\ge\theta-w(X)\), and then \(Y=K\setminus\hat X\) is such a completion.

The extension algorithm \(\mathcal{A}_{\mathrm{mw}}\), given \((I,X,\ell)\) for \(\Phi_\theta\), where the size bound \(\ell\) of the extension problem is distinct from the size \(k\) in the instance, rejects if \(|X|>k\) or \(\ell<k\); otherwise it computes \(G_X\), \(\hat X\) and \(w_X\), runs Algorithm D.10.5 on \((G_X,\hat X,w_X,k-|X|+q,\theta-w(X))\), where \(q=\operatorname{comp}(G[X])\), and if this outputs \(K\), accepts with \(Y=K\setminus\hat X\). (The input of Algorithm D.10.5 satisfies \(p=k-|X|+q\ge\max(1,q)\) because \(k\ge1\).) Its running-time exponent is \[\tag{36} g_{\mathrm{mw}}(I,X,\ell)= \begin{cases}(k-|X|)+\operatorname{comp}(G[X])&\text{if }|X|\le k\le\ell,\\0&\text{otherwise.} \end{cases}\]

Lemma D.10.7 (Extension for Maximum-Weight Connected \(k\)-Subgraph).  Let \(1\le w_{\max}\le n^{O(1)}\). The algorithm \(\mathcal{A}_{\mathrm{mw}}\) solves \(\Phi_\theta\)-Extension with one-sided error: if it accepts, \(X\cup Y\) is a connected \(k\)-set of weight at least \(\theta\) with \(Y\subseteq V(G)\setminus X\) and \(|Y|\le\ell-|X|\), and on a yes-instance it accepts with probability at least \(3/4\). It runs in polynomial space and, for every outcome of its coins, in time \(2^{g_{\mathrm{mw}}(I,X,\ell)}n^{O(1)}\).

Proof. Correctness follows from the equivalence before (36) and Lemma D.10.6. If the algorithm runs Algorithm D.10.5, then \(p=k-|X|+\operatorname{comp}(G[X])=g_{\mathrm{mw}}(I,X,\ell)\) and \(G_X\) has at most \(n\) vertices, so Lemma D.10.6 gives time \(2^{g_{\mathrm{mw}}}(nw_{\max})^{O(1)}=2^{g_{\mathrm{mw}}}n^{O(1)}\); otherwise it takes polynomial time. The running-time exponent is an integer computable in polynomial time.  ∎

Proof of Theorem 11.16. A connected \(k\)-set exists if and only if some component of \(G\) has at least \(k\) vertices; then the first \(k\) vertices of a breadth-first search order of that component form one. If none exists, report this. Otherwise, for every \(\theta\in\{-kw_{\max},\dots,kw_{\max}\}\), run Algorithm MLS-\(\theta\) for \(\Phi_\theta\) and the size \(k\) as in Corollary 11.6, with the extension algorithm \(\mathcal{A}_{\mathrm{mw}}\), \(c=2\), \(\gamma=0\), the rate \(\psi=0.71755\) of Lemma 11.12, \(\Gamma_0=16\) and \(\kappa=1\). Output the heaviest set returned by these runs, and, if no run returns a set, the connected \(k\)-set found by breadth-first search.

By Lemma D.10.7, \(\mathcal{A}_{\mathrm{mw}}\) satisfies the hypotheses of Corollary 3.15, with running-time exponent \(g_{\mathrm{mw}}\). It has a component-count bound with constants \((2,0)\): for a solution \(S\) with \(|S|=k\), let \(F\) be a spanning tree of \(G[S]\); for \(X\subseteq S\) and the size bound \(\ell=k\) we have \(|X|\le k\le\ell\), so \(g_{\mathrm{mw}}(I,X,k)=(k-|X|)+\operatorname{comp}(G[X])\le(k-|X|)\log2+|X|-e(F[X])\), because \(F[X]\) is a spanning forest of a subgraph of \(G[X]\). By Lemma 11.12, every \(\sigma\in(0,1]\) satisfies one of (R1)–(R3) with these constants. So, by Corollary 11.6, every run takes time \(2^{0.71755n}n^{O(1)}\) and polynomial space, every returned set is a solution of \(\Phi_\theta\), hence a connected \(k\)-set, and the run for \(\theta\) equal to the maximum weight returns a set with probability at least \(1-1/e\); that set has maximum weight. There are \(2kw_{\max}+1\le n^{O(1)}\) runs, so the total time is \(2^{0.71755n}n^{O(1)}\), which is \(O(1.6444^n)\) because \(0.71755<\log1.6444\) (Lemma 11.12).  ∎

D.10.2 Vertex-weighted Connected Feedback Vertex Set

Restatement of Theorem 11.17.

The weighted extension algorithm combines two ingredients. The unweighted extension algorithm (Lemma 11.7) provides a connected feedback vertex set of small excess, which serves as a structured modulator; it is only needed as a modulator, so the intermediate graphs of its iterative compression never need weights. The weighted test then runs the counting of (CC1) and (CC3) with a weight function that combines the isolating weights with the vertex weights.

The counted objects.

We use the families of the proof of Theorem A.13 of (Cygan et al. 2011), with a general weight function, and we prove the two properties of them that the weighted test needs, so that this subsection does not depend on their dynamic program. Let \(\tilde G\) be a graph with \(\tilde n\) vertices, \(Y_{\mathrm{in}}\subseteq V(\tilde G)\) nonempty and \(r\in Y_{\mathrm{in}}\), and let \(\tilde U=V(\tilde G)\times\{\mathrm F,\mathrm M\}\). For a weight function \(\mathrm{wt}\colon\tilde U\to\{1,\dots,N\}\) and integers \(a,e,c,W\), let \(\mathcal R^{a,e,c}_W\) be the family of pairs \((Z,M)\) with \(Z\subseteq V(\tilde G)\setminus Y_{\mathrm{in}}\), \(|Z|=a\), \(|E(\tilde G[Z])|=e\), \(M\subseteq Z\), \(|M|=c\) and \(\mathrm{wt}(Z\times\{\mathrm F\})+\mathrm{wt}(M\times\{\mathrm M\})=W\). Let \(\mathcal S^{a,e,c}_W\) consist of the pairs in \(\mathcal R^{a,e,c}_W\) such that \(\tilde G[Z]\) is a forest with a vertex of \(M\) in each of its components and \(\tilde G-Z\) is connected. Let \(\mathcal C^{a,e,c}_W\) consist of the triples of a pair \((Z,M)\in \mathcal R^{a,e,c}_W\), a consistent cut \((Z_1,Z_2)\) of \(\tilde G[Z]\) with \(M\subseteq Z_1\), and a consistent cut \((Y_1,Y_2)\) of \(\tilde G-Z\) with \(r\in Y_1\); a cut of a graph is consistent if no edge joins its two sides. (We write \(Z\) for the candidate forest, which (Cygan et al. 2011) call \(X\), and \(Y_{\mathrm{in}}\) for their \(S\).)

Lemma D.10.8 (Cut parity).  For every weight function \(\mathrm{wt}\) and all integers \(a,e,c,W\) with \(c\le a-e\), \(|\mathcal S^{a,e,c}_W|\equiv|\mathcal C^{a,e,c}_W|\pmod2\).

Proof. Fix \((Z,M)\in\mathcal R^{a,e,c}_W\). A cut of \(\tilde G[Z]\) is consistent exactly when it puts each component of \(\tilde G[Z]\) on one side. The components that contain a vertex of \(M\) must lie on side \(1\), and each of the \(\ell\) other components can lie on either side, so there are \(2^\ell\) cuts \((Z_1,Z_2)\). Likewise there are \(2^{\operatorname{comp}(\tilde G-Z)-1}\) cuts \((Y_1,Y_2)\), since the component of \(r\in Y_{\mathrm{in}}\subseteq V(\tilde G)\setminus Z\) must lie on side \(1\). So \((Z,M)\) lies in \(2^{\operatorname{comp}(\tilde G-Z)-1+\ell}\) triples, and this number is odd if and only if \(\tilde G-Z\) is connected and every component of \(\tilde G[Z]\) contains a vertex of \(M\). In that case \(\operatorname{comp}(\tilde G[Z])\le|M|=c\le a-e=|Z|-|E(\tilde G[Z])|\). Every graph \(H\) satisfies \(\operatorname{comp}(H)\ge|V(H)|-|E(H)|\), with equality if and only if \(H\) is a forest, so \(\tilde G[Z]\) is a forest and \((Z,M)\in\mathcal S^{a,e,c}_W\). Conversely, every pair in \(\mathcal S^{a,e,c}_W\) lies in exactly one triple. Summing over the pairs gives the claim.  ∎

A triple in \(\mathcal C^{a,e,c}_W\) is determined by its state map \(\sigma\colon V(\tilde G)\to\Sigma\), \(\Sigma=\{\mathsf y_1,\mathsf y_2,\mathsf z_1,\mathsf z_2\}\), which sends \(v\) to \(\mathsf y_j\) if \(v\in Y_j\) and to \(\mathsf z_j\) if \(v\in Z_j\), together with the marker set \(M\). A map \(\sigma\) and a set \(M\) come from a triple, in the family with the indices computed from them, if and only if

(S1)  \(\sigma(v)\in\{\mathsf y_1,\mathsf y_2\}\) for \(v\in Y_{\mathrm{in}}\), and \(\sigma(r)=\mathsf y_1\);

(S2)  no edge \(uv\) has \(\{\sigma(u),\sigma(v)\}=\{\mathsf y_1,\mathsf y_2\}\) or \(\{\sigma(u),\sigma(v)\}=\{\mathsf z_1,\mathsf z_2\}\);

(S3)  \(M\subseteq\sigma^{-1}(\mathsf z_1)\).

The indices are \(a=|\sigma^{-1}(\{\mathsf z_1,\mathsf z_2\})|\), the number \(e\) of edges with both ends in \(\sigma^{-1}(\{\mathsf z_1,\mathsf z_2\})\), \(c=|M|\) and \(W=\sum_{v\in\sigma^{-1}(\{\mathsf z_1,\mathsf z_2\})}\mathrm{wt}(v,\mathrm F)+\sum_{v\in M} \mathrm{wt}(v,\mathrm M)\). These are the four states \(0_1,0_2,1_1,1_2\) of the dynamic program of (Cygan et al. 2011) (see (CC2)). For \(\mathrm{Mod}\subseteq V(\tilde G)\) and a coloring \(\varsigma\colon\mathrm{Mod}\to\Sigma\), the core evaluation is the family of numbers \(|\mathcal C^{a,e,c}_W(\varsigma)|\bmod 2\), where \(\mathcal C^{a,e,c}_W(\varsigma)\) consists of the triples in \(\mathcal C^{a,e,c}_W\) whose state map agrees with \(\varsigma\) on \(\mathrm{Mod}\). A coloring that violates (S1) or (S2) on \(\mathrm{Mod}\) has empty families, so \(|\mathcal C^{a,e,c}_W|\) is the sum of \(|\mathcal C^{a,e,c}_W(\varsigma)|\) over the reasonable colorings \(\varsigma\) of \(\mathrm{Mod}\) (Appendix D.9.3).

Lemma D.10.9 (Weighted core evaluation).  Let \(\mathrm{wt}\colon\tilde U\to\{1,\dots,N\}\), let \(\mathrm{Mod}\subseteq V(\tilde G)\) be such that \(\tilde G-\mathrm{Mod}\) is a forest, and let \(\varsigma\colon\mathrm{Mod}\to\Sigma\). Then the numbers \(|\mathcal C^{a,e,c}_W(\varsigma)|\bmod2\) for all \(0\le a,c\le\tilde n\), \(0\le e\le\tilde n^2\) and \(0\le W\le2N\tilde n\) (all other values are \(0\)) can be computed in time and space polynomial in \(\tilde n\) and \(N\). The dynamic program has four states for each vertex of the forest, whatever the weight function; \(\mathrm{wt}\) affects only the range of the index \(W\) and the values added to it.

Proof. We compute the counts exactly and reduce them modulo \(2\). A table is an array \(T(a,e,c,W)\) of integers indexed as in the lemma, and \(T*T'\) denotes the convolution \((T*T')(a,e,c,W)=\sum T(a',e',c',W')\,T'(a-a',e-e',c-c',W-W')\), the sum over all index vectors with both arguments in range. For an index vector \(\iota\), \([\iota]\) is the table with entry \(1\) at \(\iota\) and \(0\) elsewhere, and \(T^{+e}\) is \(T\) shifted by \(1\) in the index \(e\).

Write \(Z(\sigma)=\sigma^{-1}(\{\mathsf z_1,\mathsf z_2\})\). We assign each vertex and each edge of \(\tilde G\) to one part of the computation. Let \(F=\tilde G-\mathrm{Mod}\), root each tree of \(F\) at a vertex, and let \(T_v\) be the subtree of \(F\) rooted at \(v\). Then every edge of \(\tilde G\) lies inside \(\mathrm{Mod}\), joins a forest vertex to \(\mathrm{Mod}\), or joins a forest vertex \(u\) to its parent \(v\) in \(F\). We call the last kind the parent edge of \(u\).

The modulator. If \(\varsigma\) violates (S1) or (S2) on \(\mathrm{Mod}\), all counts are \(0\). Otherwise let \(e_0\) be the number of edges of \(\tilde G[\mathrm{Mod}]\) with both ends in \(\varsigma^{-1}(\{\mathsf z_1,\mathsf z_2\})\), and put \[T_{\mathrm{Mod}}=[(0,e_0,0,0)]*\mathop{\ast}_{m\in\mathrm{Mod}}L_m,\] where \(L_m=[(0,0,0,0)]\) if \(\varsigma(m)\in\{\mathsf y_1,\mathsf y_2\}\), \(L_m=[(1,0,0,\mathrm{wt}(m,\mathrm F))]\) if \(\varsigma(m)=\mathsf z_2\), and \(L_m=[(1,0,0,\mathrm{wt}(m,\mathrm F))]+[(1,0,1,\mathrm{wt}(m,\mathrm F)+\mathrm{wt}(m,\mathrm M))]\) if \(\varsigma(m)=\mathsf z_1\). The two terms for \(\mathsf z_1\) are the choices \(m\notin M\) and \(m\in M\).

A forest vertex alone. For a vertex \(v\) of \(F\) and \(s\in\Sigma\), let \(L_v[s]=0\) if \(s\) violates (S1) at \(v\) (that is, \(v\in Y_{\mathrm{in}}\) and \(s\notin\{\mathsf y_1,\mathsf y_2\}\), or \(v=r\) and \(s\neq\mathsf y_1\)), or if some edge \(vm\) with \(m\in\mathrm{Mod}\) has \(\{s,\varsigma(m)\}=\{\mathsf y_1,\mathsf y_2\}\) or \(\{\mathsf z_1,\mathsf z_2\}\). Otherwise let \(d_v\) be the number of neighbors \(m\in\mathrm{Mod}\) of \(v\) with \(\varsigma(m)\in\{\mathsf z_1,\mathsf z_2\}\), and let \(L_v[s]\) be \([(0,0,0,0)]\) for \(s\in\{\mathsf y_1,\mathsf y_2\}\), \([(1,d_v,0,\mathrm{wt}(v,\mathrm F))]\) for \(s=\mathsf z_2\), and \([(1,d_v,0,\mathrm{wt}(v,\mathrm F))]+[(1,d_v,1,\mathrm{wt}(v,\mathrm F)+\mathrm{wt}(v,\mathrm M))]\) for \(s=\mathsf z_1\).

Subtrees. Call \(s,s'\in\Sigma\) compatible if \(\{s,s'\}\) is neither \(\{\mathsf y_1,\mathsf y_2\}\) nor \(\{\mathsf z_1,\mathsf z_2\}\). Going up from the leaves of \(F\), for each vertex \(v\) with children \(u_1,\dots,u_d\) and each \(s\in\Sigma\) put \[A_v[s]=L_v[s]*\mathop{\ast}_{i=1}^{d}\ \sum_{s'\text{ compatible with }s}A_{u_i}[s']^{(s,s')},\] where \(A^{(s,s')}=A^{+e}\) if \(s,s'\in\{\mathsf z_1,\mathsf z_2\}\) and \(A^{(s,s')}=A\) otherwise. The shift counts the parent edge of \(u_i\) when both of its ends lie in \(Z(\sigma)\). Finally, with \(\rho_1,\dots,\rho_t\) the roots of the trees of \(F\), output \[T_\varsigma=T_{\mathrm{Mod}}*\mathop{\ast}_{j=1}^{t}\ \sum_{s\in\Sigma}A_{\rho_j}[s]\] reduced modulo \(2\).

Correctness. By induction from the leaves, \(A_v[s](a,e,c,W)\) is the number of pairs \((\sigma_v,M_v)\) with the following properties. Here \(\sigma_v\colon V(T_v)\to\Sigma\) satisfies \(\sigma_v(v)=s\), satisfies (S1) on \(V(T_v)\), and satisfies (S2) on the edges inside \(T_v\) and on the edges from \(T_v\) to \(\mathrm{Mod}\), with \(\varsigma\) on \(\mathrm{Mod}\). The set \(M_v\) is a subset of \(\sigma_v^{-1}(\mathsf z_1)\). The index \(a\) is \(|Z(\sigma_v)|\), and \(e\) is the number of those edges with both ends in \(Z(\sigma_v)\cup\varsigma^{-1}(\{\mathsf z_1,\mathsf z_2\})\). The index \(c\) is \(|M_v|\), and \(W\) is the \(\mathrm{wt}\)-weight of \(Z(\sigma_v)\times\{\mathrm F\}\) and \(M_v\times\{\mathrm M\}\). The induction step holds because such a pair for \(T_v\) is exactly a choice at \(v\), counted by \(L_v[s]\), together with one such pair for each \(T_{u_i}\) whose root state is compatible with \(s\), so that (S2) holds on the parent edge of \(u_i\). All four indices add over these parts, and the parent edge of \(u_i\) is added to \(e\) exactly when both of its ends lie in \(Z(\sigma_v)\). In the same way, \(T_\varsigma\) before reduction counts the pairs \((\sigma,M)\) with \(\sigma|_{\mathrm{Mod}}=\varsigma\) that satisfy (S1)–(S3), by their indices. Each vertex of \(\tilde G\) is counted in exactly one of the parts, and so is each edge: an edge inside \(\mathrm{Mod}\) in \(e_0\), an edge from a forest vertex \(v\) to \(\mathrm{Mod}\) in \(L_v\), and a parent edge in the shift. By the correspondence between triples and pairs \((\sigma,M)\), this number is \(|\mathcal C^{a,e,c}_W(\varsigma)|\). All indices stay in range, since \(a,c\le\tilde n\), \(e\le\tilde n^2\) and \(W\le2N\tilde n\), so the truncation in the convolution loses nothing.

Time and space. A table has \((\tilde n+1)^2(\tilde n^2+1)(2N\tilde n+1)\) entries. Each entry is a nonnegative integer at most \(5^{\tilde n}\), a bound on the number of pairs \((\sigma,M)\) (each vertex has state \(\mathsf y_1\), \(\mathsf y_2\) or \(\mathsf z_2\), or state \(\mathsf z_1\) with or without a marker), so it has \(O(\tilde n)\) bits. One convolution therefore takes time polynomial in \(\tilde n\) and \(N\). The computation keeps four tables \(A_v[s]\) for each vertex whose parent has not been processed, at most \(4\tilde n\) tables in all, and performs \(O(\tilde n+|\mathrm{Mod}|)\) convolutions and sums for each of the at most \(\tilde n\) vertices. The state set \(\Sigma\), the marker choice and the recursion do not depend on \(\mathrm{wt}\). The weight function enters only through the values \(\mathrm{wt}(v,\mathrm F)\) and \(\mathrm{wt}(v,\mathrm M)\) placed in the index \(W\) by the tables \(L_m\) and \(L_v\), and through the range \(0\le W\le2N\tilde n\) of that index.  ∎

For \(N=4\tilde n\) and the weights of (CC1), Lemma D.10.9 is a direct proof of the core evaluation of (CC3). In the weighted test we use it with a weight function that also encodes the vertex weights.

Algorithm D.10.10 (Weighted test).  a graph \(\tilde G\) with \(\tilde n\) vertices, sets \(\emptyset\neq\hat X\subseteq Y_{\mathrm{in}}\subseteq V(\tilde G)\), weights \(\tilde w\colon V(\tilde G)\to\{0,\dots,w_{\max}\}\), integers \(k'\ge0\) and \(\theta'\), and a structured modulator \(\mathrm{Mod}\) for \((\tilde G,\hat X,k')\). Fix \(r\in\hat X\) and put \(\beta=8\tilde n^2+1\) and \(N'=\beta w_{\max}+4\tilde n\). For every \(\eta\) with \(|Y_{\mathrm{in}}|\le\eta\le\min(|\hat X|+k',\tilde n)\):

1.  draw \(\mathrm{iso}\colon\tilde U\to\{1,\dots,4\tilde n\}\) uniformly at random, and put \(\mathrm{wt}'(v,\mathrm F)=\beta\tilde w(v)+\mathrm{iso}(v,\mathrm F)\) and \(\mathrm{wt}'(v,\mathrm M)=\mathrm{iso}(v,\mathrm M)\), so that \(\mathrm{wt}'\) takes values in \(\{1,\dots,N'\}\);

2.  compute \(|\mathcal C^{a,e,a-e}_{W}|\bmod2\) for \(a=\tilde n-\eta\), all \(e\) and all \(W\), for the weight function \(\mathrm{wt}'\), as the sum over the reasonable colorings of \(\mathrm{Mod}\), listed by Lemma D.9.5, of the core evaluations of Lemma D.10.9;

3.  if \(|\mathcal C^{a,e,a-e}_{W}|\) is odd for some \(e\) and some \(W\) with \(\tilde w(V(\tilde G))-\lfloor W/\beta\rfloor\le\theta'\), accept.

If no \(\eta\) leads to acceptance, reject.

Lemma D.10.11 (Weighted test).  Algorithm D.10.10 runs in polynomial space and, for every outcome of its coins, in time \(3^{k'}2^{|\hat X|}(\tilde nw_{\max})^{O(1)}\). If it accepts, then \(\tilde G\) has a connected feedback vertex set \(Y\supseteq Y_{\mathrm{in}}\) with \(|Y|\le|\hat X|+k'\) and \(\tilde w(Y)\le\theta'\). If \(\tilde G\) has such a set, the algorithm accepts with probability at least \(1/2\).

Proof. For a pair \((Z,M)\) write \(\mathrm{iso}(Z,M)=\mathrm{iso}(Z\times\{\mathrm F\})+\mathrm{iso}(M\times\{\mathrm M\})\), and similarly \(\mathrm{wt}'(Z,M)\). Then \(\mathrm{wt}'(Z,M)=\beta\tilde w(Z)+\mathrm{iso}(Z,M)\) with \(0\le\mathrm{iso}(Z,M)\le2\tilde n\cdot4\tilde n<\beta\), so \(\lfloor\mathrm{wt}'(Z,M)/\beta\rfloor=\tilde w(Z)\) and \(\mathrm{wt}'(Z,M)\bmod\beta=\mathrm{iso}(Z,M)\). Fix \(\eta\) and put \(a=\tilde n-\eta\). By Lemma D.10.9 the algorithm computes the parities \(|\mathcal C^{a,e,a-e}_W|\bmod2\) for \(\mathrm{wt}'\), which equal \(|\mathcal S^{a,e,a-e}_W|\bmod2\) by Lemma D.10.8.

Soundness. Suppose the algorithm accepts at \(\eta\), \(e\) and \(W\). Then \(\mathcal S^{a,e,a-e}_W\) contains a pair \((Z,M)\). The set \(Y=V(\tilde G)\setminus Z\) contains \(Y_{\mathrm{in}}\), has size \(\eta\le|\hat X|+k'\), \(\tilde G-Y=\tilde G[Z]\) is a forest and \(\tilde G[Y]=\tilde G-Z\) is connected; so \(Y\) is a connected feedback vertex set. Its weight is \(\tilde w(Y)=\tilde w(V(\tilde G))-\tilde w(Z)=\tilde w(V(\tilde G))-\lfloor W/\beta\rfloor\le\theta'\).

Completeness. Let \(Y^*\) be a set as in the lemma and \(\eta=|Y^*|\); then \(|Y_{\mathrm{in}}|\le\eta\le \min(|\hat X|+k',\tilde n)\), so this \(\eta\) is tried. Let \(\mathcal P\) be the family of pairs \((Z,M)\) such that \(Y=V(\tilde G)\setminus Z\) is a connected feedback vertex set with \(Y\supseteq Y_{\mathrm{in}}\), \(|Y|=\eta\) and \(\tilde w(Y)\le\theta'\), and \(M\) contains exactly one vertex of each component of the forest \(\tilde G[Z]\). Identify \((Z,M)\) with \((Z\times\{\mathrm F\})\cup(M\times\{\mathrm M\})\subseteq\tilde U\); this is injective and maps \(\mathrm{iso}(Z,M)\) to the weight of the set. The family \(\mathcal P\) is nonempty, since \(Y^*\) with one marker per component of \(\tilde G-Y^*\) belongs to it. As \(|\tilde U|=2\tilde n\), the Isolation Lemma (Cygan et al. 2022 Lemma 2.5) with weights drawn from \(\{1,\dots,4\tilde n\}\) shows that with probability at least \(1/2\) a unique pair \((Z^\circ,M^\circ)\in\mathcal P\) minimizes \(\mathrm{iso}\) over \(\mathcal P\). Assume this, and put \(e^\circ=|E(\tilde G[Z^\circ])|\) and \(W^\circ=\mathrm{wt}'(Z^\circ,M^\circ)\). We claim \(\mathcal S^{a,e^\circ,a-e^\circ}_{W^\circ}=\{(Z^\circ,M^\circ)\}\). The pair \((Z^\circ,M^\circ)\) belongs to it, since \(|M^\circ|=\operatorname{comp}(\tilde G[Z^\circ])=|Z^\circ|-e^\circ\) for a forest. Let \((Z,M)\) belong to it. Then \(\tilde G[Z]\) is a forest with \(|Z|-e^\circ\) components, each containing a vertex of \(M\), and \(|M|=|Z|-e^\circ\), so each component contains exactly one vertex of \(M\); the set \(Y=V(\tilde G)\setminus Z\) is a connected feedback vertex set containing \(Y_{\mathrm{in}}\) with \(|Y|=\eta\); and \(\tilde w(Z)=\lfloor W^\circ/\beta\rfloor=\tilde w(Z^\circ)\), so \(\tilde w(Y)=\tilde w(V(\tilde G)\setminus Z^\circ)\le\theta'\). Hence \((Z,M)\in\mathcal P\) with \(\mathrm{iso}(Z,M)=W^\circ\bmod\beta= \mathrm{iso}(Z^\circ,M^\circ)\), and uniqueness gives \((Z,M)=(Z^\circ,M^\circ)\). So \(|\mathcal C^{a,e^\circ,a-e^\circ}_{W^\circ}|\) is odd, and \(\tilde w(V(\tilde G))-\lfloor W^\circ/\beta\rfloor=\tilde w(V(\tilde G)\setminus Z^\circ)\le\theta'\), so the algorithm accepts.

Time and space. There are at most \(\tilde n+1\) values of \(\eta\). By Lemma D.9.5, \(\mathrm{Mod}\) has at most \(8\cdot3^{k'}2^{|\hat X|}\) reasonable colorings, listed in that time and polynomial space; its hypotheses hold because \(\mathrm{Mod}=C\cup C'\) is a structured modulator and \(r\in\hat X\subseteq Y_{\mathrm{in}}\cap C\). By Lemma D.10.9 each core evaluation takes time and space polynomial in \(\tilde n\) and \(N'\le(8\tilde n^2+1)w_{\max}+4\tilde n\), and the parities are accumulated coloring by coloring.  ∎

We now solve the forced problem of Appendix D.9.4 with weights.

Algorithm D.10.12 (Forced weighted extension).  a forced instance \((\tilde G,\hat X,\tilde w,k',\theta')\), with \(\tilde G\) on \(\tilde n\) vertices. If \(\theta'<0\), reject.

1.  Run the algorithm \(\mathcal{A}\) of Lemma 11.7 on \((\tilde G,\hat X,|\hat X|+k')\). If it rejects, reject. Otherwise it outputs \(Y_0\); put \(A_0=\hat X\cup Y_0\) and \(\mathrm{Mod}=C\cup C'\) with \(C=A_0\) and \(C'=\emptyset\).

2.  Start with \(Y_{\mathrm{in}}=\hat X\) and go through the vertices \(v\in V(\tilde G)\setminus\hat X\) in a fixed order: run Algorithm D.10.10 on \((\tilde G,Y_{\mathrm{in}}\cup\{v\},\hat X,\tilde w,k',\theta',\mathrm{Mod})\) independently \(\lceil\log(8\tilde n)\rceil\) times, and if one run accepts, add \(v\) to \(Y_{\mathrm{in}}\).

3.  If \(Y_{\mathrm{in}}\) is a connected feedback vertex set of \(\tilde G\) with \(|Y_{\mathrm{in}}\setminus\hat X|\le k'\) and \(\tilde w(Y_{\mathrm{in}})\le\theta'\), output \(A=Y_{\mathrm{in}}\); otherwise reject.

Lemma D.10.13 (Forced weighted extension).  Algorithm D.10.12 runs in polynomial space and, for every outcome of its coins, in time \(3^{k'}2^{|\hat X|}(\tilde nw_{\max})^{O(1)}\). Every output is a solution of the input instance, and if one exists, the algorithm outputs one with probability at least \(5/8\).

Proof. The output is checked in step 3.

The modulator. If step 1 succeeds, then by Lemma 11.7 (with \(G=\tilde G\) and \(X=\hat X\)) the set \(A_0\) is a connected feedback vertex set of \(\tilde G\) with \(\hat X\subseteq A_0\) and \(|A_0\setminus\hat X|\le k'\). So \(\tilde G-\mathrm{Mod}\) is a forest, \(\tilde G[C]\) is connected, \(\hat X\subseteq C\), \(C'=\emptyset\) and \(|\mathrm{Mod}\setminus\hat X|\le k'\le k'+2\): \(\mathrm{Mod}\) is a structured modulator for \((\tilde G,\hat X,k')\), and every set \(Y_{\mathrm{in}}\cup\{v\}\) tested in step 2 contains \(\hat X\), as Algorithm D.10.10 requires.

Success. Let \(A^*\) be a solution of the instance. Then \(Y_0=A^*\setminus\hat X\) shows that \((\tilde G,\hat X,|\hat X|+k')\) is a yes-instance of Connected Feedback Vertex Set Extension, so step 1 succeeds with probability at least \(3/4\) by Lemma 11.7. Step 2 is the self-reduction of Lemma D.10.1 with \(\Omega=V(\tilde G)\), \(Z_0=\hat X\), the family \(\mathcal P\) of the connected feedback vertex sets \(Y\supseteq\hat X\) of \(\tilde G\) with \(|Y\setminus\hat X|\le k'\) and \(\tilde w(Y)\le\theta'\), which contains \(A^*\), the test \(\mathsf T(Y_{\mathrm{in}})\) given by Algorithm D.10.10 on \((\tilde G,Y_{\mathrm{in}},\hat X,\tilde w,k',\theta',\mathrm{Mod})\), \(\delta=1/8\) and \(\rho=\lceil\log(8\tilde n)\rceil\ge\log(\tilde n/\delta)\). By Lemma D.10.11, the test rejects \(Y_{\mathrm{in}}\) if no member of \(\mathcal P\) contains \(Y_{\mathrm{in}}\) (for \(Y\supseteq\hat X\) the conditions \(|Y|\le|\hat X|+k'\) and \(|Y\setminus\hat X|\le k'\) agree), and with probability at least \(1/2\) if one does. So, with probability at least \(7/8\), the final \(Y_{\mathrm{in}}\) is a member of \(\mathcal P\), and step 3 outputs it. Hence the algorithm fails with probability at most \(1/4+1/8\).

Time and space. Step 1 takes time \(3^{k'}2^{\operatorname{comp}(\tilde G[\hat X])}\tilde n^{O(1)}\le3^{k'}2^{|\hat X|}\tilde n^{O(1)}\) and polynomial space by Lemma 11.7. Step 2 runs Algorithm D.10.10 at most \(\tilde n\lceil\log(8\tilde n)\rceil\) times, each within the bounds of Lemma D.10.11.  ∎

For an instance \(I=(G,w)\) with \(G\) connected and not a forest, and an integer \(\theta\), let \(\Phi^{\mathrm{w}}_\theta\) be the implicit set system with universe \(V(G)\) whose solutions are the connected feedback vertex sets \(S\) of \(G\) with \(w(S)\le\theta\). Membership is decided in polynomial time, and every solution is nonempty.

Proof of Theorem 11.17. If \(G\) is a forest, output \(\emptyset\). If \(G\) is disconnected, then, as in Section 11.3.1, a connected feedback vertex set exists only if exactly one component \(G_1\) of \(G\) contains a cycle (as \(G\) is not a forest, at least one does), and then the connected feedback vertex sets of \(G\) are those of \(G_1\); report that none exists or replace \(G\) by \(G_1\). So let \(G\) be connected and not a forest.

Let \(\mathcal{A}_{\mathrm w}\) be the algorithm of Lemma D.9.7 with Algorithm D.10.12 as \(\mathrm{Forced}\); by Lemma D.10.13 the hypotheses of Lemma D.9.7 hold with \(p=5/8\), as \((\tilde nw_{\max})^{O(1)}\le(\tilde n+w_{\max})^{O(1)}\). So \(\mathcal{A}_{\mathrm w}\) solves \(\Phi^{\mathrm{w}}_\theta\)-Extension with one-sided error, accepts a yes-instance with probability at least \(5/8\), and runs in polynomial space and in time \(3^{k-|X|}2^{\operatorname{comp}(G[X])}n^{O(1)}\) for every outcome of its coins, because \(w_{\max}\le n^{O(1)}\). For every \(\theta\in\{1,\dots,nw_{\max}\}\) and every \(k\in\{1,\dots,n\}\), run Algorithm MLS-\(\theta\) for \(\Phi^{\mathrm{w}}_\theta\) and \(k\) as in Corollary 11.6, with the extension algorithm \(\mathcal{A}_{\mathrm w}\), \(c=3\), \(\gamma=0\), \(\psi=\log1.8\) and \(\kappa=1\). Output the lightest set returned by these runs, and \(V(G)\) if no run returns a set; \(V(G)\) is a connected feedback vertex set because \(G\) is connected.

The algorithm \(\mathcal{A}_{\mathrm w}\) satisfies the hypotheses of Corollary 3.15 with the running-time exponent \(g(G,X,k)=(k-|X|)\log3+\operatorname{comp}(G[X])\) of (22), whose comparisons with the thresholds of Algorithm MLS-\(\theta\) reduce to integer comparisons (Lemma 11.7). Every solution \(S\) induces a connected subgraph, so a spanning tree of \(G[S]\) gives the component-count bound with constants \((3,0)\), as in the proof of Theorem 11.8. By the last claim of Corollary 11.6, every run takes time \(1.8^nn^{O(1)}\) and polynomial space, and every returned set is a connected feedback vertex set of weight at most \(\theta\). Let \(S^*\) be a connected feedback vertex set of minimum weight \(\theta^*\le nw_{\max}\); the run for \(\theta=\theta^*\) and \(k=|S^*|\) returns a set with probability at least \(1-1/e\), and then the output has weight \(\theta^*\). There are \(n^2w_{\max}\le n^{O(1)}\) runs.  ∎

D.11 The further applications

This subsection contains the lemmas and proofs for Sections 12.1, 12.2, 12.3 and 12.4, the analyses of Directed Steiner Tree and Paired Domination (Appendices D.11.6 and D.11.7), and the proofs of their weighted versions.

D.11.1 Red–blue states

For a finite set \(V\) let \(V^{\mathrm b}=\{v^{\mathrm b}:v\in V\}\) and \(V^{\mathrm r}=\{v^{\mathrm r}:v\in V\}\) be two disjoint copies of \(V\), the blue and the red copy, and for \(X\subseteq V\) write \(X^{\mathrm b}=\{x^{\mathrm b}:x\in X\}\). A red–blue graph on \(V\) is a graph \(H\) with vertex set \(V^{\mathrm b}\cup V^{\mathrm r}\) in which \(v^{\mathrm b}v^{\mathrm r}\) is an edge for every \(v\in V\). A set \(S\subseteq V\) is connected red–blue dominating in \(H\) if \(S\neq\emptyset\), \(H[S^{\mathrm b}]\) is connected and every red vertex has a neighbor in \(S^{\mathrm b}\). This is the problem that Abu-Khzam, Mouawad and Liedloff (Abu-Khzam, Mouawad, and Liedloff 2011) solve in time \(O^{*}(1.3645^{|V^{\mathrm b}|+|V^{\mathrm r}|})\) and polynomial space; here it serves to formulate the extension problems of the variants as states of Definition 5.4.

Lemma D.11.1 (Red–blue states).  Let \(H\) be a red–blue graph on a set \(V\) with \(n\ge1\) elements, and for \(X\subseteq V\) put \[g_H(X)=\bigl|\{v\in V:v^{\mathrm r}\text{ has no neighbor in }X^{\mathrm b}\}\bigr|+\operatorname{comp}\bigl(H[X^{\mathrm b}]\bigr).\] Then \(g_H\) is computed in polynomial time and takes integer values in \(\{0,\dots,n\}\), with \(g_H(\emptyset)=n\) and \(g_H(S)=1\) for every connected red–blue dominating set \(S\). For every \(X\subseteq V\) and every \(\varphi\colon V\setminus X\to\{0,1,\dots,\varphi_{\max}\}\), the least weight \(\varphi(Y)\) over the sets \(Y\subseteq V\setminus X\) for which \(X\cup Y\) is connected red–blue dominating in \(H\) (\(\infty\) if there is none) is computed deterministically in time \(2^{g_H(X)}(n+\varphi_{\max})^{O(1)}\) and polynomial space; with \(\varphi\equiv1\), this decides for every integer \(k\ge0\), in time \(2^{g_H(X)}n^{O(1)}\), whether some such \(Y\) has \(|Y|\le k-|X|\). Consequently, if \(\Phi\) is an implicit set system whose instance \(I\) determines in polynomial time a red–blue graph \(H_I\) on \(U_I\) whose connected red–blue dominating sets are the solutions, then this decision algorithm solves \(\Phi\)-Extension deterministically in polynomial space, with the running-time exponent \((I,X,k)\mapsto g_{H_I}(X)\) (Definition 3.2).

Proof. Let \(\mathcal{D}\) be the family of vertex sets of the components of \(H[X^{\mathrm b}]\), let \(B=(V\setminus X)^{\mathrm b}\), and let \(R\) be the set of red vertices without a neighbor in \(X^{\mathrm b}\). Then \(J=(\mathcal{D},B,R)\) is a state of \(H\) in the sense of Definition 5.4: the members of \(\mathcal{D}\) are disjoint, nonempty and connected, no edge of \(H\) joins two distinct components of an induced subgraph, \(V(\mathcal{D})=X^{\mathrm b}\), \(B\) avoids \(X^{\mathrm b}\), and \(R\) avoids \(N_H[X^{\mathrm b}]\) because a red vertex is not in \(X^{\mathrm b}\) and the vertices of \(R\) are not adjacent to \(X^{\mathrm b}\). A subset of \(B\) has the form \(Y^{\mathrm b}\) with \(Y\subseteq V\setminus X\), and as no red vertex lies in \(Y^{\mathrm b}\), the condition \(R\subseteq N_H[Y^{\mathrm b}]\) says that every red vertex without a neighbor in \(X^{\mathrm b}\) has one in \(Y^{\mathrm b}\), that is, that every red vertex has a neighbor in \((X\cup Y)^{\mathrm b}\). So the solutions of \(J\) are exactly the sets \(Y^{\mathrm b}\) with \(Y\subseteq V\setminus X\) for which \(X\cup Y\) is connected red–blue dominating, and \(|\mathrm{T}(J)|=|R|+|\mathcal{D}|=g_H(X)\). Give the blue vertex \(y^{\mathrm b}\) of \(B\) the weight \(\varphi(y)\). By Lemma 5.6, applied to the state \(J\) of the graph \(H\) with \(2n\) vertices, the value \(\operatorname{OPT}_\varphi(J)\), which is the least weight in question, is computed deterministically in time \(2^{g_H(X)}(2n+\varphi_{\max})^{O(1)}\) and polynomial space. With \(\varphi\equiv1\) this is \(\operatorname{OPT}(J)\), the least \(|Y|\), computed in time \(2^{g_H(X)}n^{O(1)}\) by the first statement of Corollary 5.8, and one accepts if and only if \(\operatorname{OPT}(J)\le k-|X|\).

For the range, every \(v\in X\) has the neighbor \(v^{\mathrm b}\in X^{\mathrm b}\) of \(v^{\mathrm r}\), so the first term of \(g_H(X)\) is at most \(n-|X|\), and \(\operatorname{comp}(H[X^{\mathrm b}])\le|X|\). For \(X=\emptyset\) the first term is \(n\) and the second is \(0\). For a connected red–blue dominating set \(S\) the first term is \(0\) and the second is \(1\). Both terms are computed in polynomial time from \(H\) and \(X\).

For the last claim, membership in \(\mathcal{F}_I\) is tested in polynomial time through \(H_I\), so \(\Phi\) is as in Definition 3.1, and \(g_{H_I}\) is computable in polynomial time and bounds the running time of the algorithm as Definition 3.2 requires.  ∎

D.11.2 Connected \(r\)-domination

Restatement of Theorem 12.2.

Throughout, \(\mathcal{N}\) is a domination relation (Definition 12.1) on a connected graph \(G\) with \(n\ge2\) vertices.

The Steiner extension.

For \(X\subseteq V\) put \[g_\mathcal{N}(X)=|\operatorname{und}_\mathcal{N}(X)|+\operatorname{comp}(G[X]).\] For \(\mathcal{N}[v]=N[v]\) this is \(|\mathrm{T}(J_X)|\) by (13). The instance \(I\) consists of \(G\) and the description of \(\mathcal{N}\), which are fixed, and \(g_\mathcal{N}(X)\) does not depend on \(k\); in the notation of Definition 3.2, the running-time exponent is \(g(I,X,k)=g_\mathcal{N}(X)\). The next lemma shows that the Steiner extension transfers with this running-time exponent.

Lemma D.11.2 (Steiner extension for domination relations).  Let \(H_\mathcal{N}\) be the red–blue graph on \(V\) with the edges \(u^{\mathrm b}v^{\mathrm b}\) for the edges \(uv\) of \(G\) and the edges \(u^{\mathrm b}v^{\mathrm r}\) for all \(u,v\) with \(u\in\mathcal{N}[v]\). The connected red–blue dominating sets of \(H_\mathcal{N}\) are the connected \(\mathcal{N}\)-dominating sets of \(G\), and \(g_{H_\mathcal{N}}=g_\mathcal{N}\). So Lemma D.11.1 gives a deterministic extension algorithm for the implicit set system of connected \(\mathcal{N}\)-dominating sets, in time \(2^{g_\mathcal{N}(X)}n^{O(1)}\) and polynomial space, with the running-time exponent \((I,X,k)\mapsto g_\mathcal{N}(X)\), which takes values in \(\{0,\dots,n\}\).

Proof. The graph \(H_\mathcal{N}\) is built in polynomial time, because every \(\mathcal{N}[v]\) is, and \(v^{\mathrm b}v^{\mathrm r}\) is an edge since \(v\in N[v]\subseteq\mathcal{N}[v]\). For \(S\subseteq V\) the graph \(H_\mathcal{N}[S^{\mathrm b}]\) is a copy of \(G[S]\), and \(v^{\mathrm r}\) has a neighbor in \(S^{\mathrm b}\) if and only if \(\mathcal{N}[v]\cap S\neq\emptyset\). So \(S\) is connected red–blue dominating in \(H_\mathcal{N}\) if and only if \(S\) is a connected \(\mathcal{N}\)-dominating set, and \(g_{H_\mathcal{N}}(X)=g_\mathcal{N}(X)\) for every \(X\subseteq V\). Now Lemma D.11.1 applies.  ∎

Anchored trees.

The analyses of Sections 5.3 and 7.2 assign every vertex outside the solution to a neighbor inside it. For a domination relation we assign it to a solution vertex that dominates it, which need not be a neighbor, and we treat the assignment as a pendant edge of a virtual tree. In this tree the solution is a connected dominating set, and the running-time exponent of a sample is at most its number of terminals in the tree. For a graph \(T'\) on the vertex set \(V\) and \(X\subseteq V\) we write \[g_{T'}(X)=n-|N_{T'}[X]|+\operatorname{comp}(T'[X])\] for the number of terminals \(|\mathrm{T}(J_X)|\) of the state \(J_X\) of (12) formed in the graph \(T'\), by (13).

Lemma D.11.3 (Anchored tree).  Let \(S\) be a connected \(\mathcal{N}\)-dominating set, let \(F\) be a spanning tree of \(G[S]\), and let \(o:V\setminus S\to S\) be an anchor map, that is, \(o(v)\in\mathcal{N}[v]\cap S\) for every \(v\notin S\). Let \(T'\) be the graph on \(V\) formed by \(F\) and the edges \(v\,o(v)\) for \(v\notin S\). Then \(T'\) is a tree, \(S\) is a connected dominating set of \(T'\), every vertex outside \(S\) is a leaf of \(T'\), and \[g_\mathcal{N}(X)\ \le\ g_{T'}(X)\qquad\text{for every }X\subseteq S .\] Consequently, in the notation of Definition 3.3, the exponential moments \(\mu_t(S)\) for the running-time exponent \(g_\mathcal{N}\) and \(\mu^{T'}_t(S)\) for the running-time exponent \(g_{T'}\) of \(\Phi_{\mathrm{CDS}}\) on the graph \(T'\) satisfy \(\mu_t(S)\ge\mu^{T'}_t(S)\) termwise, and every integer certificate (Theorem 9.10) for \(S\) with respect to \(g_{T'}\) is one with respect to \(g_\mathcal{N}\).

Proof. The graph \(T'\) is the tree \(F\) with the \(n-|S|\) vertices outside \(S\) attached as pendant vertices, so it is a tree, \(T'[S]=F\) is connected, every \(v\notin S\) has exactly one neighbor, namely \(o(v)\in S\), and an anchor map exists because \(S\) is \(\mathcal{N}\)-dominating. Let \(X\subseteq S\). We show \(\operatorname{und}_\mathcal{N}(X)\subseteq V\setminus N_{T'}[X]\). Let \(v\in N_{T'}[X]\). If \(v\in X\), then \(v\in N[v]\subseteq\mathcal{N}[v]\) meets \(X\). Otherwise \(v\) is joined in \(T'\) to some \(x\in X\subseteq S\); if the edge \(vx\) belongs to \(F\), then \(x\in N[v]\subseteq\mathcal{N}[v]\), and if it is a pendant edge, then \(v\notin S\) and \(x=o(v)\in\mathcal{N}[v]\), because the only pendant edge at \(v\) is \(v\,o(v)\). In both cases \(v\) is dominated by \(X\). Finally \(\operatorname{comp}(G[X])\le\operatorname{comp}(F[X])=\operatorname{comp}(T'[X])\), because \(F[X]\) is a spanning subgraph of \(G[X]\) and \(T'[X]=F[X]\) for \(X\subseteq S\). Adding the two bounds gives \(g_\mathcal{N}(X)\le|V\setminus N_{T'}[X]|+\operatorname{comp}(T'[X])=g_{T'}(X)\). The last claims follow from \(2^{-g_\mathcal{N}(X)}\ge2^{-g_{T'}(X)}\) for every \(X\subseteq S\), and from \(g_\mathcal{N}(X)\le g_{T'}(X)\le\alpha-w(X)\) for every nonempty \(X\subseteq S\) when \(w\) is an integer certificate for \(S\) with respect to \(g_{T'}\).  ∎

Proof of Theorem 12.2. Let \(\Phi^\mathcal{N}\) be the implicit set system of connected \(\mathcal{N}\)-dominating sets, with the extension algorithm and the running-time exponent \(g_\mathcal{N}\) of Lemma D.11.2; \(G\) has a solution, and \(\gamma_{\mathrm{c}}^\mathcal{N}(G)\) is the least solution size. Let \(S\) be a minimum connected \(\mathcal{N}\)-dominating set, \(s^*=|S|\), and let \(T'\) be an anchored tree of Lemma D.11.3 for \(S\).

(a) Let \(\mathrm{Dec}(G,s)\) be Algorithm MLS-\(\theta\) for \(\Phi^\mathcal{N}\), with \(k=s\), \(\psi=\psi^*\), \(\Gamma=2\) and confidence \(\kappa\), that is, Algorithm RS-\(\theta\) (Algorithm 7.1) with the extension algorithm of Lemma D.11.2 in place of \(\mathcal{A}_{\mathrm{St}}\) and \(g_\mathcal{N}\) in place of \(|\mathrm{T}(J_X)|\). By Theorem 3.5, it takes time \(2^{\psi^*n}n^{O(1)}\) and polynomial space, and it rejects if \(\gamma_{\mathrm{c}}^\mathcal{N}(G)>s\). As \(T'\) is a connected graph and \(S\) is a connected dominating set of \(T'\), Proposition 7.8 applied to \(T'\) gives \(\max_t\mu^{T'}_t(S)/\binom nt\ge2^{-\psi^*n}/2\), and by Lemma D.11.3 the set \(S\) satisfies condition (\(\star\)) for \(g_\mathcal{N}\); so \(\mathrm{Dec}(G,s^*)\) accepts with probability at least \(1-e^{-\kappa}\), by Theorem 3.5. Now Lemma 5.2 gives the claim; the bound \(2^{\psi^*}<1.7204\) is as in Theorem 7.2.

(b) Run the procedure \(\mathrm{Det}\) of Lemma 10.3 for \(\Phi^\mathcal{N}\), the extension algorithm and running-time exponent of Lemma D.11.2 and \((\alpha,\Xi)=(n-1,n-2)\), for \(s=1,\dots,n\), and output the least \(s\) at which it answers yes. By Lemma 10.3, it answers no if \(\gamma_{\mathrm{c}}^\mathcal{N}(G)>s\). At \(s=s^*\) the set \(S\) has size exactly \(s\), and it has an integer certificate for \((n-1,n-2)\): the tree \(T'\) is a spanning tree of the graph \(T'\) in which every vertex outside \(S\) is a leaf, so the weights \(w(u)=d_{T'}(u)-1\) on \(S\) form the certificate (19) for \(S\) in the graph \(T'\), that is, they are nonnegative integers with \(w(S)=n-2\) and \(g_{T'}(X)\le n-1-w(X)\) for every nonempty \(X\subseteq S\), by Lemma 5.10 and (13) applied to \(T'\), and by Lemma D.11.3 they form an integer certificate for \(S\) with respect to \(g_\mathcal{N}\). So \(\mathrm{Det}\) answers yes at \(s=s^*\), and Lemma 5.2, in its deterministic form and with the time and space bounds of Lemma 10.3, gives the theorem.  ∎

D.11.3 Vertex-weighted connected domination

Restatement of Theorem 12.3.

The notation is that of Section 12.2.

Lemma D.11.4 (Weighted max leaf and connected domination).  If \(n\ge3\), the maximum leaf weight of a spanning tree of \(G\) equals \(\varphi(V)-\gamma_{\mathrm{c}}^\varphi(G)\).

Proof. If \(F\) is a spanning tree, then \(V\setminus\operatorname{leaves}(F)\) is a connected dominating set, by the first half of the proof of Lemma 5.1, so \(\varphi(\operatorname{leaves}(F))=\varphi(V)-\varphi(V\setminus\operatorname{leaves}(F))\le \varphi(V)-\gamma_{\mathrm{c}}^\varphi(G)\). Conversely, for a connected dominating set \(S\) the second half of that proof gives a spanning tree in which every vertex of \(V\setminus S\) is a leaf; as \(\varphi\ge0\), its leaf weight is at least \(\varphi(V\setminus S)=\varphi(V)-\varphi(S)\).  ∎

The weighted Steiner extension.

For \(X\subseteq V\) let \[\operatorname{OPT}_\varphi(X)=\min\{\varphi(Y):Y\subseteq V\setminus X,\ X\cup Y\text{ is a connected dominating set}\}.\] Since the solutions of the state \(J_X\) of (12) are the sets \(Y\subseteq V\setminus X\) for which \(X\cup Y\) is a connected dominating set (also for \(X=\emptyset\)), this is \(\operatorname{OPT}_\varphi(J_X)\) in the notation of Definition 5.5, with \(\varphi\) restricted to the blue vertices \(V\setminus X\), and Lemma 5.6 computes it deterministically in time \(2^{|\mathrm{T}(J_X)|}(n+\varphi_{\max})^{O(1)}\) and polynomial space.

Relaxed extension.

The weighted extension does not control the size of the completion: Lemma 5.3 minimizes weight, and a completion of least weight may be large. We show that this does not affect Theorems 3.5 and 9.10: their proofs use the correctness of the extension algorithm in only two places. The following lemma extends the two theorems accordingly.

Lemma D.11.5 (Relaxed extension algorithms).  Let \(\Phi\) be an implicit set system, and let \(\mathcal{A}\) be a deterministic algorithm on inputs \((I,X,k)\) such that

(R1)  \(\mathcal{A}\) accepts \((I,X,k)\) whenever some \(Y\subseteq U_I\setminus X\) with \(|Y|\le k-|X|\) has \(X\cup Y\in\mathcal{F}_I\), and

(R2)  \(\mathcal{A}\) rejects \((I,X,k)\) if \(\mathcal{F}_I=\emptyset\).

Let \(g\) satisfy Definition 3.2 for \(\mathcal{A}\). Then Theorem 3.5, with its ranges for the cap and the trial counts, and Theorem 9.10 hold for \(\mathcal{A}\) and \(g\), with the soundness statement “answers no if \(\mathcal{F}_I\) contains no set of size at most \(k\)” replaced by “answers no if \(\mathcal{F}_I=\emptyset\).”

Proof. An algorithm for \(\Phi\)-Extension satisfies (R1) and (R2), and the proofs of the two theorems use the correctness of \(\mathcal{A}\) only through these two properties, in the following places. The soundness parts (item (ii) of Theorem 3.5, and the paragraph called soundness in the proof of Theorem 9.10) conclude from an accepting run of \(\mathcal{A}\) that a solution exists; by (R2), \(\mathcal{F}_I\neq\emptyset\). The completeness parts (item (iii) of Theorem 3.5, and the paragraph called completeness) run \(\mathcal{A}\) on \((I,X,k)\) with \(X\subseteq S\), \(S\in\mathcal{F}_I\) and \(|S|=k\), and conclude that it accepts because \(Y=S\setminus X\) is a witness; this is (R1). The statements about time and space use only that \(\mathcal{A}\) runs in time \(2^{g(I,X,k)}N^{O(1)}\) on every input \((I,X,k)\), and the early answer of Algorithm 9.9, the rounding arguments, Lemma 3.8, Theorem 9.6 and Lemma 9.7 do not involve \(\mathcal{A}\).  ∎

Budget loop.

The weighted problems of this paper are solved by running a decision algorithm for every weight budget and every solution size. The following lemma states this wrapper once.

Lemma D.11.6 (Budget loop).  Let \(\Phi\) be an implicit set system whose instances \(I\) include a weight function \(w\colon U_I\to\{0,1,\dots,w_{\max}\}\), written in unary, and write \(w(S)=\sum_{u\in S}w(u)\). For an integer \(\varpi\ge0\) let \(\Phi_\varpi\) be the implicit set system with the same instances and universes whose solutions are the solutions \(S\) of \(\Phi\) with \(w(S)\le\varpi\). Let \(p\in(0,1]\), and suppose that for every \(\varpi\) an algorithm \(\mathcal{B}_\varpi\), given \(I\) and an integer \(k\in\{0,\dots,n\}\), answers yes or no within time \(T\) and polynomial space, such that

(B1)  \(\mathcal{B}_\varpi(I,k)\) answers no if \(\Phi_\varpi\) has no solution on \(I\), and

(B2)  if \(\Phi\) has a solution on \(I\), then some solution \(S^*\) of minimum weight has the property that \(\mathcal{B}_{w(S^*)}(I,|S^*|)\) answers yes with probability at least \(p\).

Let the algorithm \(\mathcal{B}\) run \(\mathcal{B}_\varpi(I,k)\) for every \(\varpi\in\{0,\dots,w(U_I)\}\) and every \(k\in\{0,\dots,n\}\), output the least \(\varpi\) for which some run answers yes, and report that \(\Phi\) has no solution on \(I\) if no run answers yes. Then \(\mathcal{B}\) runs in time \((w(U_I)+1)(n+1)T+N^{O(1)}\) and polynomial space, it never outputs a value below the minimum weight of a solution, and with probability at least \(p\) it outputs the minimum weight of a solution (respectively reports correctly that there is none).

Proof. If \(\Phi\) has no solution on \(I\), then no \(\Phi_\varpi\) has one, so by (B1) no run answers yes and the report is correct. Otherwise let \(S^*\) be as in (B2) and \(\varpi^*=w(S^*)\). Every solution has weight at least \(\varpi^*\), so for \(\varpi<\varpi^*\) the system \(\Phi_\varpi\) has no solution, and by (B1) no run for such a \(\varpi\) answers yes; hence the output, if any, is at least \(\varpi^*\). The run \(\mathcal{B}_{\varpi^*}(I,|S^*|)\) is among the runs, because \(\varpi^*\le w(U_I)\) and \(|S^*|\le n\), and it answers yes with probability at least \(p\) by (B2); then the output is \(\varpi^*\). There are \((w(U_I)+1)(n+1)\) runs, and \(w(U_I)\le N\) because \(w\) is written in unary.  ∎

In the applications, \(\mathcal{B}_\varpi\) is Algorithm RS-\(\theta\), Algorithm 9.9 or Algorithm MLS-\(\theta\) for \(\Phi_\varpi\), with a relaxed extension algorithm (Lemma D.11.5) or an extension algorithm with one-sided error (Corollary 3.15); property (B1) is the soundness statement of these theorems, and property (B2) is their completeness at \(k=|S^*|\), from condition (\(\star\)) with \(p=1-e^{-\kappa}\) or from an integer certificate with \(p=1\); the completeness is used only at the size of \(S^*\).

The algorithms.

For a budget \(\varpi\in\{0,\dots,\varphi(V)\}\) let \(\Phi_{\mathrm{wCDS}}\) be the implicit set system whose instances are triples \((G,\varphi,\varpi)\), with universe \(V\) and with the connected dominating sets \(S\) with \(\varphi(S)\le\varpi\) as solutions. Let \(\mathcal{A}_\varphi\) be the algorithm that, on \(((G,\varphi,\varpi),X,k)\), computes \(\operatorname{OPT}_\varphi(X)\) by Lemma 5.6 and accepts if and only if \(\varphi(X)+\operatorname{OPT}_\varphi(X)\le\varpi\); it ignores \(k\). It satisfies (R1) and (R2): a witness \(Y\) in (R1) gives \(\varphi(X)+\operatorname{OPT}_\varphi(X)\le \varphi(X\cup Y)\le\varpi\), and an acceptance yields a connected dominating set \(X\cup Y\) of weight at most \(\varpi\). By Lemma 5.6, the function \(g\) with \(g((G,\varphi,\varpi),X,k)=|\mathrm{T}(J_X)|\) is a running-time exponent for \(\mathcal{A}_\varphi\) in the sense of Definition 3.2, with integer values in \(\{0,\dots,n\}\); it depends neither on \(k\) nor on \(\varphi\) and \(\varpi\), and in the shorthand declared after Definition 3.2 it is \(g(X)=|\mathrm{T}(J_X)|\).

Proof of Theorem 12.3. For \(n\le2\) the maximum leaf weight is output directly, as stated in the theorem, since \(G\) is its own only spanning tree. For \(n\ge3\), by Lemma D.11.4 it suffices to compute \(\gamma_{\mathrm{c}}^\varphi(G)\), which we do by Lemma D.11.6 for the implicit set system of connected dominating sets with the weights \(\varphi\), so that \(\Phi_\varpi\) is \(\Phi_{\mathrm{wCDS}}\) on \((G,\varphi,\varpi)\) and \(w(U_I)=\varphi(V)\le n\varphi_{\max}\). In (a), \(\mathcal{B}_\varpi(G,\varphi,s)\) is the run for the candidate size \(s\) of Theorem 7.2 (Algorithm RS-\(\theta\), Algorithm 7.1) for \(\Phi_{\mathrm{wCDS}}\) on \((G,\varphi,\varpi)\), with the relaxed extension algorithm \(\mathcal{A}_\varphi\) in place of \(\mathcal{A}_{\mathrm{St}}\) and the running-time exponent \(|\mathrm{T}(J_X)|\). In (b), \(\mathcal{B}_\varpi(G,\varphi,s)\) is the procedure \(\mathrm{Det}\) of Lemma 10.3 for \(\Phi_{\mathrm{wCDS}}\) on \((G,\varphi,\varpi)\) with \(\mathcal{A}_\varphi\), its running-time exponent \(|\mathrm{T}(J_X)|\) and \((\alpha,\Xi)=(n-1,n-2)\): by Lemma D.11.5, Theorem 9.10 holds for \(\mathcal{A}_\varphi\) with the soundness statement “only if \(\Phi_{\mathrm{wCDS}}\) has a solution,” so Lemma 10.3, whose proof uses \(\mathcal{A}_\varphi\) only through Theorem 9.10 and Lemma 10.2, holds for \(\mathcal{A}_\varphi\) with the same change; its enumeration tests membership in \(\Phi_{\mathrm{wCDS}}\), that is, connected domination and weight at most \(\varpi\), in polynomial time per set.

Property (B1). By Lemma D.11.5 with Theorem 3.5 in (a), and by Lemma 10.3 as just stated in (b), \(\mathcal{B}_\varpi(G,\varphi,s)\) answers no if \(\Phi_{\mathrm{wCDS}}\) has no solution on \((G,\varphi,\varpi)\), that is, if \(G\) has no connected dominating set of weight at most \(\varpi\).

Property (B2). Let \(S^*\) be a connected dominating set with \(\varphi(S^*)=\gamma_{\mathrm{c}}^\varphi(G)\), and put \(s^*=|S^*|\) and \(\varpi^*=\varphi(S^*)\); \(S^*\) is a solution of \(\Phi_{\mathrm{wCDS}}\) on \((G,\varphi,\varpi^*)\) of size exactly \(s^*\). In (a), Proposition 7.8 gives condition (\(\star\)) for \(S^*\) with the running-time exponent \(|\mathrm{T}(J_X)|\), which does not involve the weights, so by Lemma D.11.5 with Theorem 3.5(iii) the run \(\mathcal{B}_{\varpi^*}(G,\varphi,s^*)\) answers yes with probability at least \(1-e^{-\kappa}\). In (b), the certificate (19) for \(S^*\) is an integer certificate for \((n-1,n-2)\), so \(\mathcal{B}_{\varpi^*}(G,\varphi,s^*)\) answers yes by Lemma 10.3.

Time and space. By Lemma D.11.5, each run in (a) has the time and space bounds of the corresponding run in Theorem 7.2, and each run in (b) those of Lemma 10.3, with \(N^{O(1)}=(n+\varphi_{\max})^{O(1)}\). So Lemma D.11.6 gives the claims, with \(p=1-e^{-\kappa}\) in (a) and \(p=1\) in (b), and a factor \((n\varphi_{\max}+1)(n+1)\) in the time.  ∎

D.11.4 Weakly connected domination

Restatement of Theorem 12.4.

The notation is that of Section 12.3. For \(X\subseteq V\) the running-time exponent is \[\tag{37} g_{\mathrm w}(X)=n-|N[X]|+\operatorname{comp}(G^2[X]),\] the analogue of (13) with \(G^2[X]\) in place of \(G[X]\).

Lemma D.11.7 (Characterization).  A set \(S\subseteq V\) is a weakly connected dominating set of \(G\) if and only if \(S\) is dominating and \(G^2[S]\) is connected.

Proof. Let \(S\) be a weakly connected dominating set, and let \(x,x'\in S\). The graph \(\langle S\rangle_{\mathrm w}\) is connected, so it contains a path from \(x\) to \(x'\). A vertex \(v\notin S\) on this path is an end of its two path edges, each of which has an end in \(S\); so both path neighbors of \(v\) lie in \(S\). Hence any two consecutive vertices of \(S\) on the path are adjacent in \(G\) or have a common neighbor, that is, they are adjacent in \(G^2[S]\), and \(x\) and \(x'\) lie in one component of \(G^2[S]\).

Conversely, let \(S\) be dominating with \(G^2[S]\) connected. Then \(\langle S\rangle_{\mathrm w}\) has vertex set \(N[S]=V\). Every \(v\notin S\) has a neighbor \(u\in S\), and \(uv\) is an edge of \(\langle S\rangle_{\mathrm w}\). If \(x,x'\in S\) are adjacent in \(G^2[S]\), then \(xx'\) is an edge of \(\langle S\rangle_{\mathrm w}\), or \(x\) and \(x'\) have a common neighbor \(u\) and \(xu\), \(ux'\) are edges of \(\langle S\rangle_{\mathrm w}\). So all of \(S\) lies in one component of \(\langle S\rangle_{\mathrm w}\), and every other vertex is adjacent to \(S\) in it.  ∎

By Lemma D.11.7 and the graph \(H_{\mathrm w}\) of Lemma D.11.8 below, the weakly connected dominating sets of \(G\) are the connected red–blue dominating sets of \(H_{\mathrm w}\), so the algorithm of Abu-Khzam, Mouawad and Liedloff (Abu-Khzam, Mouawad, and Liedloff 2011) for Connected Red-Blue Dominating Set solves Weakly Connected Dominating Set in time \(O^{*}(1.3645^{2n})=O^{*}(1.8619^{n})\) and polynomial space.

Lemma D.11.8 (Extension for weak connectivity).  Let \(H_{\mathrm w}\) be the red–blue graph on \(V\) with the edges \(u^{\mathrm b}v^{\mathrm b}\) for all \(u\neq v\) at distance at most \(2\) in \(G\) and the edges \(u^{\mathrm b}v^{\mathrm r}\) for all \(u,v\) with \(u\in N[v]\). The connected red–blue dominating sets of \(H_{\mathrm w}\) are the weakly connected dominating sets of \(G\), and \(g_{H_{\mathrm w}}=g_{\mathrm w}\). So Lemma D.11.1 gives a deterministic extension algorithm for the implicit set system of weakly connected dominating sets, in time \(2^{g_{\mathrm w}(X)}n^{O(1)}\) and polynomial space, with the running-time exponent \((G,X,k)\mapsto g_{\mathrm w}(X)\), which takes values in \(\{0,\dots,n\}\).

Proof. The graph \(H_{\mathrm w}\) is built in polynomial time. For \(S\subseteq V\) the graph \(H_{\mathrm w}[S^{\mathrm b}]\) is a copy of \(G^2[S]\), and \(v^{\mathrm r}\) has a neighbor in \(S^{\mathrm b}\) if and only if \(N[v]\cap S\neq\emptyset\); so by Lemma D.11.7 the set \(S\) is connected red–blue dominating in \(H_{\mathrm w}\) if and only if it is weakly connected dominating, and \(g_{H_{\mathrm w}}(X)=n-|N[X]|+\operatorname{comp}(G^2[X])=g_{\mathrm w}(X)\) by (37). Now Lemma D.11.1 applies, for the implicit set system of weakly connected dominating sets.  ∎

The next lemma is the analogue of Lemma 5.10; unlike (19), it also holds for the empty sample.

Lemma D.11.9 (Pointwise certificate for weak connectivity).  Let \(S\) be a weakly connected dominating set, let \(T\) be a spanning tree of \(\langle S\rangle_{\mathrm w}\), rooted anywhere, and for \(u\in S\) let \(w(u)\) be \(d_T(u)\) minus the number of children of \(u\) in \(T\) that lie in \(S\). Then \(w\) takes nonnegative integer values, \(w(S)=n-1\), and \[g_{\mathrm w}(X)\ \le\ n-w(X)\qquad\text{for every }X\subseteq S .\]

Proof. Every edge of \(\langle S\rangle_{\mathrm w}\) has an end in \(S\), so \(S\) is a vertex cover of \(T\). Charge every edge of \(T\), joining a vertex \(v\) to its parent \(p\), to \(v\) if \(v\in S\) and to \(p\) otherwise; in the second case \(p\in S\), because \(S\) covers the edge. A vertex \(u\in S\) is charged with the edge to its parent (if \(u\) is not the root) and with the edges to its children outside \(S\), that is, with exactly \(w(u)\ge0\) edges. So \(w(S)=|E(T)|=n-1\). Let \(E_T(X)\) be the set of edges of \(T\) with an end in \(X\). An edge charged to a vertex of \(X\) has that vertex as an end, so \(w(X)\le|E_T(X)|\).

It remains to show \(g_{\mathrm w}(X)\le n-|E_T(X)|\). Let \(T_X\) be the graph with vertex set \(N_T[X]\) and edge set \(E_T(X)\); every edge of \(E_T(X)\) has both ends in \(N_T[X]\), and \(T_X\) is a forest, being a subgraph of \(T\), so it has \(|N_T[X]|-|E_T(X)|\) components. Let \(x,x'\in X\) lie in one component of \(T_X\), and take a path of \(T_X\) from \(x\) to \(x'\). A vertex \(v\notin X\) on this path is an end of its two path edges, each of which has an end in \(X\); so both path neighbors of \(v\) lie in \(X\). Hence consecutive vertices of \(X\) on the path are at distance at most \(2\) in \(T\), and so in \(G\), that is, they are adjacent in \(G^2[X]\); thus \(x\) and \(x'\) lie in one component of \(G^2[X]\). Therefore \(\operatorname{comp}(G^2[X])\le\operatorname{comp}(T_X)=|N_T[X]|-|E_T(X)|\), and with \(N_T[X]\subseteq N[X]\), as \(T\subseteq G\), \[\begin{aligned} g_{\mathrm w}(X)&=n-|N[X]|+\operatorname{comp}(G^2[X])\le n-|N_T[X]|+|N_T[X]|-|E_T(X)|\\ &=n-|E_T(X)|\le n-w(X). \end{aligned}\]  ∎

Proof of Theorem 12.4. Run the procedure \(\mathrm{Det}\) of Lemma 10.3 for the implicit set system of weakly connected dominating sets, the extension algorithm and running-time exponent \(g_{\mathrm w}\) of Lemma D.11.8 and \((\alpha,\Xi)=(n,n-1)\), for \(s=1,\dots,n\), and output the least \(s\) at which it answers yes; \(V\) is a solution, and \(\gamma_{\mathrm w}(G)\) is the least solution size. By Lemma 10.3, \(\mathrm{Det}\) answers no if \(\gamma_{\mathrm w}(G)>s\). At \(s=\gamma_{\mathrm w}(G)\) a minimum weakly connected dominating set \(S\) has size exactly \(s\), and the weights of Lemma D.11.9 are an integer certificate for \(S\) and \((n,n-1)\): they are nonnegative integers with \(w(S)=n-1=\Xi\) and \(g_{\mathrm w}(X)\le\alpha-w(X)\) for every \(X\subseteq S\). So \(\mathrm{Det}\) answers yes at \(s=\gamma_{\mathrm w}(G)\), and Lemma 5.2, in its deterministic form and with the bounds of Lemma 10.3, gives the theorem.  ∎

Unlike for Connected Dominating Set, we do not expect the exponential moments of the running-time exponent \(g_{\mathrm w}\) to give a smaller running time than its affine certificate: on the following instances the inequality of Lemma D.11.9 is an equality. For integers \(s\ge2\) and \(d_1,\dots,d_s\ge2\), the spider \(T\) is the tree with a center \(z\) adjacent to vertices \(v_1,\dots,v_s\), where \(v_i\) carries \(d_i-1\) pendant vertices; so \(T\) has \(n=1+\sum_id_i\) vertices.

Proposition D.11.10 (Spiders).  Let \(s\ge2\) and \(d_1,\dots,d_s\ge2\) be integers, let \(T\) be the spider with these parameters, and let \(S=\{v_1,\dots,v_s\}\). Then:

(a)  \(S\) is the unique minimum weakly connected dominating set of \(T\).

(b)  \(g_{\mathrm w}(X)=n-\sum_{v_i\in X}d_i\) for every \(X\subseteq S\), so the generating function of Definition 3.3 for the running-time exponent \(g_{\mathrm w}\) is \(W_S(y)=2^{-n}\prod_{i=1}^s\bigl(1+y2^{d_i}\bigr)\).

Proof. (a) The sets \(\{v_i\}\cup \operatorname{pend}(v_i)\), where \(\operatorname{pend}(v_i)\) is the set of pendant vertices of \(v_i\), are disjoint, and each meets every weakly connected dominating set, since a pendant vertex must be dominated. So a set of size at most \(s\) has exactly one vertex in each of them and does not contain \(z\). If such a set \(S'\) contains a pendant vertex of \(v_i\) instead of \(v_i\), then no edge at \(v_i\) other than those to \(\operatorname{pend}(v_i)\) has an end in \(S'\), so \(\{v_i\}\cup \operatorname{pend}(v_i)\) is a union of components of \(\langle S'\rangle_{\mathrm w}\) that misses \(z\), and \(\langle S'\rangle_{\mathrm w}\) is disconnected. Hence \(S'=S\). And \(S\) dominates \(T\) and every edge of \(T\) has an end in \(S\), so \(\langle S\rangle_{\mathrm w}=T\) is connected.

(b) For \(X=\emptyset\) both sides equal \(n\). For nonempty \(X\subseteq S\), \(N[X]\) consists of \(z\) and the sets \(\{v_i\}\cup \operatorname{pend}(v_i)\) with \(v_i\in X\), so \(|N[X]|=1+\sum_{v_i\in X}d_i\), and \(G^2[X]\) is connected, as any two vertices of \(X\) have the common neighbor \(z\). Now (37) gives the formula, and the product form of \(W_S\) follows by expanding the product over the subsets of \(S\).  ∎

By Proposition D.11.10(b), on spiders \(g_{\mathrm w}\) equals the integer affine surrogate exponent \(n-w(X)\) of Lemma D.11.9 for \(T\) rooted at \(z\), whose weights \(d_i\in\{2,3\}\) form the worst integer profile of Lemma 9.7 when \(s/\Xi\in[\frac13,\frac12]\). Let \(\beta_s\) be the largest value of \(\bigl(\inf_{y>0}(1+y)^n/W_S(y)\bigr)^{1/n}\) over \(d_1,\dots,d_s\in\{2,3\}\). Numerical computation (Appendix F.5) gives values of \(\beta_s\) that appear to decrease toward \(\sqrt5-\frac12\); this is not proved. By Proposition D.11.10(a), Lemma 3.9 and Proposition 3.10, if \(\beta_s\) tends to \(\sqrt5-\frac12\), then Algorithm MLS-\(\theta\) with the running-time exponent \(g_{\mathrm w}\) needs time \((\sqrt5-\frac12)^{n-o(n)}\) on spiders and cannot improve on Theorem 12.4 by more than a subexponential factor.

D.11.5 Directed Max Leaf Spanning Tree

Restatement of Theorem 12.5.

The notation is that of Section 12.4. Write \(N^-(v)\) for the set of in-neighbors of \(v\), and \(\mathrm{ch}_F(x)\) for the set of children of \(x\) in an out-branching \(F\).

Lemma D.11.11 (Internal sets).  Let \(r\in V(\vec G)\). Then:

(a)  The set of non-leaves of an out-branching rooted at \(r\) is \(r\)-internal.

(b)  From an \(r\)-internal set \(S\) one computes in polynomial time an out-branching rooted at \(r\) in which every vertex outside \(S\) is a leaf.

(c)  Every superset of an \(r\)-internal set is \(r\)-internal.

Hence the maximum number of leaves of an out-branching is \(n-\min|S|\), the minimum over all \(r\) and all \(r\)-internal sets \(S\), and an out-branching rooted at \(r\) exists if and only if \(V(\vec G)\) is \(r\)-internal.

Proof. (a) Let \(F\) be an out-branching rooted at \(r\) and \(S\) its set of non-leaves. As \(n\ge2\), \(r\) has a child, so \(r\in S\). For \(v\in S\), every vertex of the path of \(F\) from \(r\) to \(v\) has a child or is \(v\), so the path lies in \(\vec G[S]\). A leaf \(v\) is not \(r\), and its parent is a non-leaf and an in-neighbor of \(v\). (b) Take a breadth-first search out-tree of \(\vec G[S]\) from \(r\), which spans \(S\), and attach every \(v\notin S\) by an arc from an in-neighbor in \(S\). (c) If \(S\subseteq S'\), then a vertex of \(S'\setminus S\) has an in-neighbor in \(S\), which is reachable from \(r\) in \(\vec G[S]\subseteq \vec G[S']\); the other conditions are inherited. The final statements follow from (a) and (b), and from (c) applied to \(S'=V(\vec G)\).  ∎

For a fixed root \(r\), let \(\Phi_r\) be the implicit set system whose instances are pairs \((\vec G,r)\), with universe \(V(\vec G)\) and the \(r\)-internal sets as solutions. For \(X\subseteq V(\vec G)\) let \[\operatorname{und}_r(X)=\{v\notin X\cup\{r\}:N^-(v)\cap X=\emptyset\},\] let \(\mathrm{Src}_r(X)\) be the set of strong components \(C\) of \(\vec G[X]\) that are sources (no arc of \(\vec G[X]\) enters \(C\) from \(X\setminus C\)) and do not contain \(r\), and put \[g_r(X)=|\operatorname{und}_r(X)|+|\mathrm{Src}_r(X)| .\] The members of \(\mathrm{Src}_r(X)\) are disjoint, and each contains a vertex of \(X\setminus\{r\}\), so \(g_r(X)\le(n-|X\cup\{r\}|)+|X\setminus\{r\}|=n-1\); also \(g_r(\emptyset)=n-1\), and \(g_r(S)=0\) for every \(r\)-internal set \(S\), because \(\operatorname{und}_r(S)=\emptyset\) and the strong component of \(r\) is the only source of \(\vec G[S]\).

The extension problem for \(\Phi_r\) is an instance of Directed Steiner Tree, on a digraph with a blue and a red copy of every vertex, and Lemma 12.7 solves it. Let \(\vec G^{\,r}\) be the digraph with a vertex \(v^{\mathrm b}\) for every \(v\in V(\vec G)\) and a vertex \(v^{\mathrm r}\) for every \(v\neq r\), with the arcs \(u^{\mathrm b}v^{\mathrm b}\) for every arc \(uv\) of \(\vec G\), the arcs \(u^{\mathrm b}v^{\mathrm r}\) for every arc \(uv\) of \(\vec G\) with \(v\neq r\), and the arcs \(v^{\mathrm b}v^{\mathrm r}\) for \(v\neq r\). We consider the instance \((\vec G^{\,r},r^{\mathrm b},K)\) of Directed Steiner Tree with root \(r^{\mathrm b}\) and terminals \(K=\{v^{\mathrm r}:v\neq r\}\); its non-terminals are \(U=\{v^{\mathrm b}:v\neq r\}\), and we write \(g_{\mathrm{DST}}\) for the running-time exponent \(g\) of Section 12.5 on this instance. Red vertices have no outgoing arcs. So for \(S\subseteq V(\vec G)\) with \(r\in S\), in the digraph \(\vec G^{\,r}[\{r^{\mathrm b}\}\cup K\cup(S\setminus\{r\})^{\mathrm b}]\) the blue vertices reachable from \(r^{\mathrm b}\) are the copies of the vertices reachable from \(r\) in \(\vec G[S]\), and a red vertex \(v^{\mathrm r}\) is reachable if and only if one of its in-neighbors in this digraph, which are \(v^{\mathrm b}\) if \(v\in S\) and \(u^{\mathrm b}\) for \(u\in S\cap N^-(v)\), is reachable. Hence \((S\setminus\{r\})^{\mathrm b}\) is a rooted Steiner set of \((\vec G^{\,r},r^{\mathrm b},K)\) if and only if \(S\) is \(r\)-internal: the blue vertices of \(S\setminus\{r\}\) are reachable if and only if every vertex of \(S\) is reachable from \(r\) in \(\vec G[S]\), and then a red vertex \(v^{\mathrm r}\) is reachable if and only if \(v\in S\) or \(v\) has an in-neighbor in \(S\).

Lemma D.11.12 (Directed extension).  For every \(X\subseteq V(\vec G)\) and integer \(k\ge0\) one can decide in time \(2^{g_r(X)}n^{O(1)}\) and polynomial space, deterministically, whether some \(Y\subseteq V(\vec G)\setminus X\) with \(|Y|\le k-|X|\) makes \(X\cup Y\) an \(r\)-internal set. So \(((\vec G,r),X,k)\mapsto g_r(X)\) is a running-time exponent (Definition 3.2) for \(\Phi_r\), with values in \(\{0,\dots,n-1\}\).

Proof. Put \(X'=(X\setminus\{r\})^{\mathrm b}\subseteq U\) and \(k'=k-|X|-[r\notin X]\); the root \(r\) belongs to every \(r\)-internal set, so if \(r\notin X\) it must be added, using one of the \(k-|X|\) further elements. If \(k'<0\), reject. Otherwise run the algorithm of Lemma 12.7 on the instance \((\vec G^{\,r},r^{\mathrm b},K)\) with the set \(X'\) and the size bound \(|X'|+k'\), and return its answer. This is correct: the sets \(Y\subseteq V(\vec G)\setminus X\) with \(|Y|\le k-|X|\) for which \(X\cup Y\) is \(r\)-internal correspond, by \(Y\mapsto Y'=(Y\setminus\{r\})^{\mathrm b}\), to the sets \(Y'\subseteq U\setminus X'\) with \(|Y'|\le k'\) for which \(X'\cup Y'\) is a rooted Steiner set. Indeed, if \(X\cup Y\) is \(r\)-internal, then \(r\in X\cup Y\), so \(r\in Y\) exactly when \(r\notin X\), \(|Y'|=|Y|-[r\notin X]\), and \((X\cup Y\setminus\{r\})^{\mathrm b}=X'\cup Y'\) is rooted by the equivalence above; conversely, for \(Y'\subseteq U\setminus X'\) put \(Y=\{v:v^{\mathrm b}\in Y'\}\cup(\{r\}\setminus X)\), so that \(|Y|=|Y'|+[r\notin X]\), \(r\in X\cup Y\) and \((X\cup Y\setminus\{r\})^{\mathrm b}=X'\cup Y'\), and \(X\cup Y\) is \(r\)-internal when \(X'\cup Y'\) is rooted.

The algorithm of Lemma 12.7 runs in time \(2^{g_{\mathrm{DST}}(X')}n^{O(1)}\) and polynomial space, as \(\vec G^{\,r}\) has \(2n-1\) vertices. We show \(g_{\mathrm{DST}}(X')\le g_r(X)\). Let \(W=\{r^{\mathrm b}\}\cup K\cup X'\), whose blue vertices are the copies of \(X\cup\{r\}\), and let \(W^{r}\) be the set of vertices reachable from \(r^{\mathrm b}\) in \(\vec G^{\,r}[W]\). Let \(A\) be the set of vertices reachable from \(r\) in \(\vec G[X\cup\{r\}]\); since red vertices have no outgoing arcs, the blue vertices of \(W^{r}\) are exactly \(A^{\mathrm b}\), and the blue vertices of \(W\setminus W^{r}\) are \((X\setminus A)^{\mathrm b}\), which induce a copy of \(\vec G[X\setminus A]\). Let \(C\) be a source component of \(\vec G^{\,r}[W\setminus W^{r}]\); we assign to it an element of \(\operatorname{und}_r(X)\) or of \(\mathrm{Src}_r(X)\), injectively.

If \(C\) contains a red vertex \(v^{\mathrm r}\), then \(C=\{v^{\mathrm r}\}\), because every vertex of a strong component with at least two vertices has an outgoing arc inside it. Let \(u^{\mathrm b}\) be an in-neighbor of \(v^{\mathrm r}\) in \(\vec G^{\,r}\) with \(u\in X\cup\{r\}\), so that \(u^{\mathrm b}\in W\). If \(u^{\mathrm b}\in W^{r}\), then \(v^{\mathrm r}\in W^{r}\), and otherwise the arc \(u^{\mathrm b}v^{\mathrm r}\) enters \(C\) from \((W\setminus W^{r})\setminus C\); both contradict the choice of \(C\). So no in-neighbor of \(v^{\mathrm r}\) has its underlying vertex in \(X\cup\{r\}\): \(v\notin X\) (as \(v^{\mathrm b}\) is an in-neighbor), \(v\neq r\), and \(N^-(v)\cap X=\emptyset\). Hence \(v\in\operatorname{und}_r(X)\).

Otherwise \(C\) consists of blue vertices, \(C=D^{\mathrm b}\) with \(D\subseteq X\setminus A\), and \(D\) is strongly connected in \(\vec G[X\setminus A]\). No arc of \(\vec G[X]\) enters \(D\) from \(X\setminus D\): an arc from \(X\setminus(A\cup D)\) would enter \(C\) from \((W\setminus W^{r})\setminus C\) in \(\vec G^{\,r}[W\setminus W^{r}]\), and an arc from a vertex of \(A\) to a vertex \(d\in D\) would make \(d\) reachable from \(r\) in \(\vec G[X\cup\{r\}]\), that is, \(d\in A\). So \(D\) is a strong component of \(\vec G[X]\) (a vertex of \(X\setminus D\) in the same strong component would have a path to \(D\) in \(\vec G[X]\), whose last arc enters \(D\)), it is a source, and \(r\notin D\) because \(r\in A\). Hence \(D\in\mathrm{Src}_r(X)\).

Distinct source components give distinct elements, so \(g_{\mathrm{DST}}(X')\le|\operatorname{und}_r(X)|+|\mathrm{Src}_r(X)|=g_r(X)\). The value \(g_r(X)\), and membership in \(\Phi_r\), are computed in polynomial time, and the range was shown before the lemma.  ∎

The proof of the certificate is the directed analogue of Lemma D.11.14: the source components of \(\vec G[X]\) are bounded by the vertices of \(X\) whose parent is not in \(X\).

Lemma D.11.13 (Pointwise certificate for out-branchings).  Let \(S\) be an \(r\)-internal set, and let \(F\) be an out-branching rooted at \(r\) in which every vertex outside \(S\) is a leaf (Lemma D.11.11(b)). Then for every \(X\subseteq S\), \[g_r(X)\ \le\ n-1-\sum_{x\in X}|\mathrm{ch}_F(x)| .\] Consequently the weights \(w(x)=|\mathrm{ch}_F(x)|-[x=r]\) on \(S\) are nonnegative integers with \(w(S)=n-2\), and \(g_r(X)\le n-1-w(X)\) for every \(X\subseteq S\); this is the certificate (19).

Proof. Let \(Q\) be the set of vertices \(v\neq r\) whose parent in \(F\) is not in \(X\). Every vertex other than \(r\) has exactly one parent, and the children sets \(\mathrm{ch}_F(x)\) are disjoint, so \(|Q|=n-1-\sum_{x\in X}|\mathrm{ch}_F(x)|\). We show that \(\operatorname{und}_r(X)\subseteq Q\setminus X\) and that \(\mathrm{Src}_r(X)\) maps injectively into \(Q\cap X\); adding the two bounds gives \(g_r(X)\le|Q|\).

A vertex \(v\in\operatorname{und}_r(X)\) is not \(r\) and not in \(X\), and its parent is an in-neighbor of \(v\), so the parent is not in \(X\); hence \(v\in Q\setminus X\). Let \(C\in\mathrm{Src}_r(X)\), let \(x\in C\), and let \(u\) be the ancestor of \(x\) in \(F\) closest to \(r\) such that the path of \(F\) from \(u\) to \(x\) lies in \(X\). This path is a path of \(\vec G[X]\) ending in \(C\), and no arc of \(\vec G[X]\) enters the source component \(C\) from outside, so \(u\in C\). As \(r\notin C\), \(u\neq r\), and by the choice of \(u\) its parent is not in \(X\); so \(u\in Q\cap X\cap C\). As the members of \(\mathrm{Src}_r(X)\) are disjoint, choosing such a \(u\) for each of them is injective.

Every vertex other than \(r\) has a parent, which is a non-leaf and so lies in \(S\); hence \(\sum_{x\in S}|\mathrm{ch}_F(x)|=n-1\) and \(w(S)=n-2\). As \(n\ge2\), \(r\) has a child, so \(w\ge0\), and \(w(X)\le\sum_{x\in X}|\mathrm{ch}_F(x)|\) gives \(g_r(X)\le n-1-w(X)\).  ∎

Proof of Theorem 12.5. For \(n=1\) the answer is \(1\); let \(n\ge2\). For every \(r\) for which \(V(\vec G)\) is \(r\)-internal (by Lemma D.11.11, the other roots carry no out-branching), compute the least size \(s_r\) of an \(r\)-internal set, and output \(n-\min_rs_r\); if there is no such \(r\), report that there is no out-branching. This is correct by Lemma D.11.11.

To compute \(s_r\), run the procedure \(\mathrm{Det}\) of Lemma 10.3 for \(\Phi_r\), the extension algorithm and running-time exponent \(g_r\) of Lemma D.11.12 and \((\alpha,\Xi)=(n-1,n-2)\), for \(s=1,\dots,n\), and let \(s_r\) be the least \(s\) at which it answers yes; \(V(\vec G)\) is a solution of \(\Phi_r\), and \(s_r\) is the least solution size. By Lemma 10.3, \(\mathrm{Det}\) answers no if \(s_r>s\). At \(s=s_r\) a minimum \(r\)-internal set \(S\) has size exactly \(s\), and Lemma D.11.13 gives its integer certificate for \((n-1,n-2)\), so \(\mathrm{Det}\) answers yes. Hence Lemma 5.2, in its deterministic form, computes \(s_r\) within the time and space bounds of Lemma 10.3, and running the computation for each of the at most \(n\) roots multiplies the running time by at most \(n\).  ∎

D.11.6 Directed Steiner Tree

This subsection proves Theorem 12.8, with the notation of Section 12.5. Fix an inclusion-minimal rooted Steiner set \(S\), put \(s=|S|\) and \(p=|K|\), and let \(A\) be an out-arborescence of \(\vec G[\{r\}\cup K\cup S]\) rooted at \(r\); it exists because \(S\) is rooted. By minimality every leaf of \(A\) other than \(r\) is a terminal: if a vertex \(v\in S\) were a leaf, \(A-v\) would show that \(S\setminus\{v\}\) is rooted. For \(u\neq r\) write \(\mathrm{par}(u)\) for the parent of \(u\) in \(A\). The trees of the forest \(A[\bar K_X]\) that do not contain \(r\) bound the running-time exponent \(g(X)\) of a sample \(X\subseteq S\) from above.

Lemma D.11.14 (Pointwise bound).  For every \(X\subseteq S\), \[g(X)\ \le\ \Psi_A(X):=\bigl|\{u\in K\cup X:\ \mathrm{par}(u)\notin \bar K_X\}\bigr| .\] Equality holds if \(\vec G[\bar K_X]=A[\bar K_X]\).

Lean proof: fa_dst_point, fa_dst_point_eq, fa_dst_W (formalisation)

Proof. The forest \(A[\bar K_X]\) consists of out-trees; the tree containing \(r\) lies in \(\bar K^{r}_X\), and every other tree has a root \(u\in K\cup X\) with \(\mathrm{par}(u)\notin \bar K_X\). So at most \(\Psi_A(X)\) trees miss \(r\). A tree whose root lies in \(\bar K^{r}_X\) lies entirely in \(\bar K^{r}_X\), as its arcs are arcs of \(\vec G[\bar K_X]\). So every \(v\in \bar K_X\setminus \bar K^{r}_X\) lies in a tree with root \(u\in \bar K_X\setminus \bar K^{r}_X\), and the tree path from \(u\) to \(v\) avoids \(\bar K^{r}_X\); hence \(v\) is reachable from \(u\) within \(\vec G[\bar K_X\setminus \bar K^{r}_X]\). If \(v\) lies in a source component \(C\), then so does \(u\), since nothing outside \(C\) reaches \(C\). Assigning to each source component the root of the tree of one of its vertices is therefore injective, and \(g(X)\le\Psi_A(X)\). If \(\vec G[\bar K_X]=A[\bar K_X]\), then \(\vec G[\bar K_X\setminus \bar K^{r}_X]\) is a forest of out-trees whose source components are exactly the roots counted by \(\Psi_A\).  ∎

Consequently, with the quantities of Definition 3.3 for the universe \(U\), \[\tag{38} W_S(y)=\sum_{X\subseteq S}y^{|X|}2^{-g(X)}\ \ge\ W_A(y):=\sum_{X\subseteq S}y^{|X|}2^{-\Psi_A(X)} \qquad(y\ge0).\] Whether a vertex \(u\) is counted in \(\Psi_A(X)\) depends only on whether \(u\) and \(\mathrm{par}(u)\) lie in \(\bar K_X\). This locality lets us compute \(W_A\) by a recursion along \(A\) (Lemma D.11.15) with two states: a vertex lies in \(\bar K_X\) (a terminal, the root, or a sampled vertex), or it is an unsampled vertex of \(S\). For \(x,x'\in\mathbb{R}^2\) let \[x\circ x'=\Bigl(x_1(x'_1+x'_2),\ x_2\bigl(\tfrac{x'_1}2+x'_2\bigr)\Bigr),\] a bilinear map with nonnegative coefficients. For a vertex \(v\) of \(A\) let \(A_v\) be the subtree of \(v\) and its descendants. For \(y\ge0\) we define vectors \(\mathrm{msg}_v\in\mathbb{R}^2\) for the vertices \(v\) of \(A\), from the leaves up. Put \(\mathrm{Init}_v=(1,0)\) if \(v\in K\cup\{r\}\) and \(\mathrm{Init}_v=(y,1)\) if \(v\in S\), and for \(v\) with children \(o_1,\dots,o_\ell\), in any order, let \(\mathrm{msg}_v=\bigl(\cdots(\mathrm{Init}_v\circ\mathrm{msg}_{o_1})\circ\cdots\bigr)\circ\mathrm{msg}_{o_\ell}\).

Lemma D.11.15 (Two-state recursion).  Let \(y\ge0\), and let the vectors \(\mathrm{msg}_v\) be defined with this \(y\) and any order of the children. Then \(W_A(y)=(\mathrm{msg}_r)_1\).

Lean proof: dst_rec (formalisation)

Proof. For a vertex \(v\) and \(X\subseteq S\cap V(A_v)\) put \(\Psi_v(X)=|\{u\in V(A_v)\setminus\{v\}:u\in K\cup X,\ \mathrm{par}(u)\notin \bar K_X\}|\); this depends only on \(X\), as \(\mathrm{par}(u)\in V(A_v)\) for these \(u\). Say that \(X\) puts \(v\) in state \(1\) if \(v\in \bar K_X\) and in state \(2\) otherwise. We show by induction that \((\mathrm{msg}_v)_i\) is the sum of \(y^{|X|}2^{-\Psi_v(X)}\) over the sets \(X\subseteq S\cap V(A_v)\) that put \(v\) in state \(i\); more precisely, that the same holds after the first \(j\) children, for the subtree \(A_v^j\) formed by \(v\) and the subtrees of \(o_1,\dots,o_j\), and for \(\Psi_v^j\) defined like \(\Psi_v\) on \(A_v^j\).

For \(j=0\) the subtree is \(\{v\}\) and \(\Psi_v^0=0\). If \(v\in K\cup\{r\}\), the only set is \(\emptyset\), in state \(1\), giving \((1,0)\); if \(v\in S\), the set \(\{v\}\) is in state \(1\) with weight \(y\) and \(\emptyset\) in state \(2\) with weight \(1\), giving \((y,1)\). For the step to \(j\), write \(o=o_j\) and split \(X=X'\cup X_o\) with \(X'\subseteq V(A_v^{j-1})\) and \(X_o\subseteq V(A_o)\). Then \(\Psi_v^j(X)=\Psi_v^{j-1}(X')+\Psi_o(X_o)+[o\in \bar K_X\text{ and }v\notin \bar K_X]\), because \(o\) is the only vertex counted in \(\Psi_v^j\) whose parent lies in the other part. Summing the weights, a state-\(1\) vertex \(v\) receives the factor \((\mathrm{msg}_o)_1+(\mathrm{msg}_o)_2\) and a state-\(2\) vertex \(v\) the factor \(\frac12(\mathrm{msg}_o)_1+(\mathrm{msg}_o)_2\), which is the product \(\circ\). Finally \(r\in \bar K_X\) always and \(r\) is not counted, so \(W_A(y)=\sum_Xy^{|X|}2^{-\Psi_r(X)}=(\mathrm{msg}_r)_1\).  ∎

Antinorm certificates.

As for RS-\(\theta\) (Lemma D.4.2), a region of the plane that \(\circ\) maps into itself, up to known scale factors, bounds \(W_A\) from below for all arborescences at once. With two states the region lies in the plane, and the cone added to the convex hull can be the nonnegative quadrant. For a finite set \(Z\subseteq\mathbb{R}^2_{\ge0}\) whose points have positive first coordinate, let \(\mathcal R_Z=\mathrm{conv}(Z)+\mathbb{R}^2_{\ge0}\) and \(c_Z=\min_{z\in Z}z_1\). Minimality of \(S\) enters through the vertices of \(S\): since none of them is a leaf of \(A\), their initial vector \(\mathrm{Init}_v=(y,1)\) is never the vector \(\mathrm{msg}_v\) of a complete subtree, and it is needed only as a left factor of \(\circ\) (condition (C3) below).

Lemma D.11.16 (Certificate).  Let \(y\ge0\), \(e_1,e_2>0\), and let \(Z\subseteq\mathbb{R}^2_{\ge0}\) be a finite set whose points have positive first coordinate, with

(C1)  \(z\circ z'\in\mathcal R_Z\) for all \(z,z'\in Z\);

(C2)  \((1,0)\in e_2\mathcal R_Z\);

(C3)  \((y,1)\circ z\in e_1\mathcal R_Z\) for all \(z\in Z\).

Then \(W_A(y)\ge c_Z\,e_1^{\,s}e_2^{\,p+1}\) for every out-arborescence \(A\) as above, that is, rooted at \(r\), with \(s\) vertices in \(S\), \(p\) terminals, and no leaf in \(S\).

Lean proof: dst_tree_bound_WA, dst_tree_bound (formalisation)

Proof. First, if \(x\in\eta\mathcal R_Z\) and \(x'\in\eta'\mathcal R_Z\) with \(\eta,\eta'>0\), then \(x\circ x'\in\eta\eta'\mathcal R_Z\), and \((y,1)\circ x'\in e_1\eta'\mathcal R_Z\). Indeed, write \(x=\eta(z+q)\) and \(x'=\eta'(z'+q')\) with \(z,z'\in\mathrm{conv}(Z)\) and \(q,q'\in\mathbb{R}^2_{\ge0}\). By bilinearity, \(x\circ x'=\eta\eta'\bigl(z\circ z'+z\circ q'+q\circ(z'+q')\bigr)\). Writing \(z\) and \(z'\) as convex combinations of points of \(Z\) shows that \(z\circ z'\) is a convex combination of products of points of \(Z\), so it lies in the convex set \(\mathcal R_Z\) by (C1); the other two terms are nonnegative, because \(\circ\) maps pairs of nonnegative vectors to nonnegative vectors, and \(\mathcal R_Z+\mathbb{R}^2_{\ge0}=\mathcal R_Z\). Likewise \((y,1)\circ x'=\eta'\bigl((y,1)\circ z'+(y,1)\circ q'\bigr)\) with \((y,1)\circ z'\in e_1\mathcal R_Z\) by (C3) and convexity, and \((y,1)\circ q'\ge0\).

Let \(s_v\) and \(p_v\) be the numbers of vertices of \(A_v\) in \(S\) and in \(K\cup\{r\}\). We show \(\mathrm{msg}_v\in e_1^{\,s_v}e_2^{\,p_v}\mathcal R_Z\) by induction from the leaves. If \(v\in K\cup\{r\}\), then \(\mathrm{Init}_v=(1,0)\in e_2\mathcal R_Z\) by (C2), and each child \(o\) multiplies the scale by that of \(\mathrm{msg}_o\), by the first claim. If \(v\in S\), then \(v\) is not a leaf; with its first child \(o_1\), the second claim gives \(\mathrm{Init}_v\circ\mathrm{msg}_{o_1}\in e_1e_1^{\,s_{o_1}}e_2^{\,p_{o_1}}\mathcal R_Z\), and the remaining children are handled by the first claim. At the root, \(s_r=s\) and \(p_r=p+1\). Every point of \(\mathcal R_Z\) has first coordinate at least \(c_Z\), so Lemma D.11.15 gives \(W_A(y)=(\mathrm{msg}_r)_1\ge c_Ze_1^{\,s}e_2^{\,p+1}\).  ∎

Proposition D.11.17 (Certificates for Directed Steiner Tree).  The certificates of Appendix F.7 are three rational certificates \((y_j,e_{1,j},e_{2,j},Z_j)\), \(j=0,1,2\), each of which satisfies (C1)–(C3) with \(y=y_j\), \(e_1=e_{1,j}\), \(e_2=e_{2,j}\) and \(Z=Z_j\), and rational weights \(\lambda_j\ge0\) with \(\sum_j\lambda_j=1\). With the exponent vectors \(\boldsymbol\alpha_j=(\alpha_{j,1},\alpha_{j,2},\alpha_{j,3})=\bigl(\log(1+y_j)-\log e_{1,j},\ -\log e_{2,j},\ \log(1+y_j)\bigr)\), the data are, approximately,

\(j\)	\(\lambda_j\approx\)	\(y_j\approx\)	\(e_{1,j}\approx\)	\(e_{2,j}\approx\)	\(|Z_j|\)	\(c_{Z_j}e_{2,j}\ge\)	\((\alpha_{j,1},\alpha_{j,2},\alpha_{j,3})\approx\)
\(0\)	\(0.7458\)	\(0.45494\)	\(1.003883\)	\(0.78\)	\(27\)	\(0.40\)	\((0.535367,\,0.358454,\,0.540958)\)
\(1\)	\(0.0858\)	\(0.48223\)	\(1.041122\)	\(0.78\)	\(10\)	\(0.41\)	\((0.509635,\,0.358454,\,0.567774)\)
\(2\)	\(0.1684\)	\(0.02\)	\(0.963015\)	\(0.52\)	\(28\)	\(0.011\)	\((0.082938,\,0.943416,\,0.028569)\)

and \[\max_{i\in\{1,2,3\}}\ \sum_{j=0}^2\lambda_j\alpha_{j,i}\ =\ 0.4569669\ldots\ <\ 0.457 .\]

Lean proof: dst_certs_exact, checkCert_sound, dst_rates (formalisation) · Certificates: dst_certificates.json, dst_certificate_verify.py, cz_bounds.py

Proof. The hypotheses (C1)–(C3), the displayed maximum, which lies more than \(3.3\cdot10^{-5}\) below \(0.457\), and the inequality \(0.457<\log1.3727\) are checked in Appendix F.7.  ∎

Restatement of Theorem 12.8.

Proof. Let \(\psi=0.457\); we may assume \(n_U\ge1\). Run Algorithm MLS-\(\theta\) for \(\Phi_{\mathrm{DST}}\), the extension algorithm of Lemma 12.7 and the running-time exponent \(g\), with universe size \(n_U\), rate \(\psi n/n_U\), \(\Gamma=100\) and confidence \(\kappa\), for \(k=0,1,\dots,n_U\), and output the least accepted \(k\), or report that no Steiner set exists if no \(k\) is accepted. By Theorem 3.5 and Remark 3.6, this takes time \(2^{\psi n}n^{O(1)}\) and polynomial space, and no accepted \(k\) is below the optimum; in particular, no \(k\) is accepted if there is no Steiner set. Let \(S\) be a minimum Steiner set; it is rooted and inclusion-minimal, so for \(k=|S|\) it remains to verify condition (\(\star\)). Let \(r_0=n_U-s\), so that \(s+p+r_0=n-1\). For each \(j\), equation (38) and Lemma D.11.16 and the first inequality of Lemma 3.9 give \[\max_{0\le t\le s}\frac{\mu_t(S)}{\binom{n_U}t}\ \ge\ \frac{W_S(y_j)}{(1+y_j)^{n_U}} \ \ge\ c_{Z_j}e_{2,j}\,\frac{e_{1,j}^{\,s}e_{2,j}^{\,p}}{(1+y_j)^{s+r_0}} \ =\ c_{Z_j}e_{2,j}\,2^{-(\alpha_{j,1}s+\alpha_{j,2}p+\alpha_{j,3}r_0)} .\] Choose \(j\) with the least exponent. By Proposition D.11.17, \(\alpha_{j,1}s+\alpha_{j,2}p+\alpha_{j,3}r_0\le\sum_i\lambda_i(\alpha_{i,1}s+\alpha_{i,2}p+\alpha_{i,3}r_0)\le\psi(n-1)\), and \(c_{Z_j}e_{2,j}\ge1/100\). So \(\max_t\mu_t(S)/\binom{n_U}t\ge2^{-(\psi n/n_U)n_U}/\Gamma\), which is (\(\star\)). Finally \(2^{0.457}=1.37268\ldots\).  ∎

D.11.7 Paired Domination

This subsection proves Lemma 12.9, Theorem 12.10 and Proposition 12.11, with the notation of Section 12.6.

Restatement of Lemma 12.9.

Proof. Fix a target size \(s\in\{|X|,\dots,\min(n,k)\}\) and a target weight \[b\in\bigl\{0,\dots,\lfloor\min(\varpi,w(V))\rfloor\bigr\};\] there are at most \((n+1)(nw_{\max}+1)\) such pairs. A set \(A\supseteq X\) dominates \(G\) if and only if it dominates every vertex of \(\operatorname{und}(X)\), that is, \(A\cap N[u]\neq\emptyset\) for all \(u\in \operatorname{und}(X)\). For \(F\subseteq \operatorname{und}(X)\) let \(V_F=V\setminus N[F]\), and consider the polynomial \[\mathsf F_{s,b}=\sum_{F\subseteq \operatorname{und}(X)}(-1)^{|F|}\sum_{\substack{X\subseteq A\subseteq V_F\\|A|=s,\ w(A)=b}}\det\mathbf T_A .\] Exchanging the sums, the coefficient of \(\det\mathbf T_A\), for \(A\supseteq X\) with \(|A|=s\) and \(w(A)=b\), is \[\sum_{F\subseteq \operatorname{und}(X)}(-1)^{|F|}\prod_{u\in F}[A\cap N[u]=\emptyset] =\prod_{u\in \operatorname{und}(X)}\bigl(1-[A\cap N[u]=\emptyset]\bigr),\] which is \(1\) if \(A\) dominates \(\operatorname{und}(X)\) and \(0\) otherwise. So \[\mathsf F_{s,b}=\sum_{A}\mathrm{Pf}(\mathbf T_A)^2,\] summed over the dominating sets \(A\supseteq X\) of size \(s\) and weight \(b\).

No cancellation. Let \(M\) be a perfect matching of \(G[A]\) for such an \(A\). In \(\mathrm{Pf}(\mathbf T_{A'})^2=\sum_{M_1,M_2}\pm x_{M_1}x_{M_2}\), a product \(x_{M_1}x_{M_2}\) equals \(x_M^2\) only if every edge of \(M\) lies in both \(M_1\) and \(M_2\) and no other edge occurs, that is, \(M_1=M_2=M\) and \(A'=V(M)=A\); this term has coefficient \((\pm1)^2=1\). So the coefficient of \(x_M^2\) in \(\mathsf F_{s,b}\) is exactly \(1\), and \(\mathsf F_{s,b}\neq0\) if and only if some dominating \(A\supseteq X\) of size \(s\) and weight \(b\) has a perfect matching in \(G[A]\), that is, if \(A\setminus X\) is a completion of size \(s-|X|\) with \(w(A)=b\).

Evaluation. For \(F\subseteq \operatorname{und}(X)\) we have \(X\subseteq V_F\), because no vertex of \(F\) lies in \(N[X]\). Let \(\mathbf E\) be the diagonal matrix on \(V_F\) with \(\mathbf E_{vv}=z^{w(v)}\) for \(v\notin X\) and \(\mathbf E_{vv}=0\) for \(v\in X\), with a further indeterminate \(z\). Expanding the determinant along the diagonal, with one more indeterminate \(x_\star\), \[\det\bigl(\mathbf T_{V_F}+x_\star\mathbf E\bigr)=\sum_{Q\subseteq V_F\setminus X}x_\star^{|Q|}z^{w(Q)} \det\mathbf T_{V_F\setminus Q},\] so the inner sum of \(\mathsf F_{s,b}\) is the coefficient of \(x_\star^{|V_F|-s}z^{w(V_F)-b}\). Choose a prime \(q\) with \(4n+nw_{\max}\le q\le8n+2nw_{\max}\), by trial division, and substitute independent uniform elements of \(\mathbb F_q\) for the \(x_e\). The determinant is then a polynomial in \(x_\star\) and \(z\) of degree at most \(n\) in \(x_\star\) and at most \(nw_{\max}<q\) in \(z\), so evaluating it at \((n+1)(nw_{\max}+1)\) points and interpolating gives the coefficient over \(\mathbb F_q\) in polynomial time, and summing over \(F\) gives \(\mathsf F_{s,b}\) at the random point, in time \(2^{|\operatorname{und}(X)|}(n+w_{\max})^{O(1)}\) and polynomial space. Accept if the value is nonzero for some \(s\) and \(b\).

If no completion of weight at most \(\varpi\) exists, every \(\mathsf F_{s,b}\) with \(b\le\varpi\) is the zero polynomial and the algorithm never accepts. If a completion of size \(s-|X|\) and weight \(b\le\varpi\) exists, then \(\mathsf F_{s,b}\) is a nonzero polynomial of degree \(s\le n\) in the \(x_e\) with a coefficient \(1\), so it is nonzero over \(\mathbb F_q\), and by the Schwartz–Zippel lemma (Schwartz 1980; Zippel 1979) it vanishes at the random point with probability at most \(n/q\le\frac14\). With \(w\equiv0\) and \(\varpi=0\) only \(b=0\) occurs, the determinant does not involve \(z\), and the bounds are polynomial in \(n\).  ∎

Fix a paired dominating set \(S\) and a perfect matching \(M\) of \(G[S]\), and choose an anchor \(\mathrm{anc}(v)\in S\cap N(v)\) for every \(v\notin S\). For \(u\in S\) let \(d_u=|\mathrm{anc}^{-1}(u)|\) be the number of vertices anchored at \(u\), and for a pair \(uv\in M\) let \(n_{uv}=2+d_u+d_v\), the number of vertices the pair is responsible for; so \(\sum_{uv\in M}n_{uv}=n\).

Lemma D.11.18 (Pointwise certificate).  For every \(X\subseteq S\), \[|\operatorname{und}(X)|\ \le\ \bigl|\{v\notin S:\mathrm{anc}(v)\notin X\}\bigr|+2\,\bigl|\{uv\in M:u,v\notin X\}\bigr| ,\] and consequently, for every \(y\ge0\), \[W_S(y)=\sum_{X\subseteq S}y^{|X|}2^{-|\operatorname{und}(X)|}\ \ge\ \prod_{uv\in M}\Omega_{uv}(y),\qquad \Omega_{uv}(y)=y^2+y\bigl(2^{-d_u}+2^{-d_v}\bigr)+2^{-n_{uv}} .\]

Lean proof: pd_point, pd_point_factor, card_eq_sum_pairs, pairing_of_matching, matching_of_pairing, exists_anchor, Ωpair_amgm (formalisation)

Proof. A vertex \(v\notin S\) is dominated once \(\mathrm{anc}(v)\in X\). A vertex \(u\in S\) is dominated once \(u\) or its partner in \(M\) lies in \(X\). This gives the first inequality. The right-hand side is a sum over the pairs of \(M\) of a term that depends only on \(X\cap\{u,v\}\): it is \(0\) if \(u,v\in X\), it is \(d_v\) if only \(u\in X\), it is \(d_u\) if only \(v\in X\), and it is \(n_{uv}\) if neither is in \(X\). So \(\sum_{X\subseteq S}y^{|X|}2^{-(\text{right-hand side})}\) factorizes over the pairs into the \(\Omega_{uv}(y)\).  ∎

By the inequality of arithmetic and geometric means, \(2^{-d_u}+2^{-d_v}\ge2\cdot2^{-(n_{uv}-2)/2}\), so \(\Omega_{uv}(y)\ge y^2+4\cdot2^{-n_{uv}/2}y+2^{-n_{uv}}\) for \(y\ge0\). The functions \(h_{\bar n}\), the point \(y^*\) and the rate \(\psi_{\mathrm{pr}}\) are those of Section 12.6.

Lemma D.11.19 (Rate).  The functions \(h_{\bar n}\) and the point \(y^*\) defined above satisfy \[\max_{\bar n\in\{2,3,\dots\}}h_{\bar n}(y^*)=h_6(y^*)=\psi_{\mathrm{pr}}=0.7050097\ldots,\] and for every graph \(G\) without isolated vertices and every paired dominating set \(S\) of \(G\), \[W_S(y^*)\ \ge\ (1+y^*)^n\,2^{-\psi_{\mathrm{pr}}n} .\]

Lean proof: pd_rate, pd_rate_W (formalisation) · Certificates: paired_rate_certificate.py

Proof. For \(\bar n\ge41\) we have \(y^2+4\cdot2^{-\bar n/2}y+2^{-\bar n}\ge y^2\) and \(0<y^*<1\), so \(h_{\bar n}(y^*)\le\log(1+y^*)+\frac2{41}\log\frac1{y^*}<0.431\). For \(2\le \bar n\le40\), \(\bar n\neq6\), direct evaluation (Appendix F.8) gives \(h_6(y^*)-h_{\bar n}(y^*)>0.00297\), the smallest gap occurring at \(\bar n=7\); it also gives \(\log(1+y^*)+\frac2{41}\log\frac1{y^*}<0.431<h_6(y^*)\), which completes the tail bound. This proves the first claim. By Lemma D.11.18 and the bound \(\Omega_{uv}(y)\ge y^2+4\cdot2^{-n_{uv}/2}y+2^{-n_{uv}}\), \[\begin{split} \log\frac{(1+y^*)^n}{W_S(y^*)}\ &\le\ \sum_{uv\in M}\Bigl(n_{uv}\log(1+y^*)-\log\Omega_{uv}(y^*)\Bigr)\\ &\le\ \sum_{uv\in M}n_{uv}\,h_{n_{uv}}(y^*)\ \le\ \psi_{\mathrm{pr}}\sum_{uv\in M}n_{uv} =\psi_{\mathrm{pr}}\,n . \end{split}\]  ∎

Restatement of Theorem 12.10.

Proof. Let \(\psi=0.70502\), a rational with \(\psi_{\mathrm{pr}}<\psi\) and \(2^\psi<1.6302\). For \(k=2,4,\dots\) up to \(n\), run Algorithm MLS-\(\theta\) for \(\Phi_{\mathrm{PD}}\), the extension algorithm of Lemma 12.9 (with \(w\equiv0\) and \(\varpi=0\)), the running-time exponent \(|\operatorname{und}(X)|\), rate \(\psi\), \(\Gamma=1\) and confidence \(\kappa\); as this extension algorithm has one-sided error, the cap and the trial counts are those of Corollary 3.15, that is, those of Theorem 3.5 for \(2\Gamma=2\). Output the least accepted \(k\), or \(n\) if no \(k\) is accepted (as in Lemma 5.2). By Theorem 3.5 with Corollary 3.15, each run takes time \(2^{\psi n}n^{O(1)}\) and polynomial space and rejects if no paired dominating set of size at most \(k\) exists. For \(k=\gamma_{\mathrm{pr}}(G)\), a minimum paired dominating set satisfies the generating-function condition of Theorem 3.5 at \(y=y^*\) with \(\Gamma=1\), by Lemma D.11.19, so by Corollary 3.15 this \(k\) is accepted with probability at least \(1-e^{-\kappa}\).  ∎

Restatement of Proposition 12.11.

Proof. The derivative of \(h_6\) vanishes exactly where \(\frac1{1+y}=\frac{2y+1/2}{6(y^2+y/2+1/64)}\), that is, where \(4y^2+\frac y2-\frac{13}{32}=0\); the only positive root is \(y^*\). As \(h_6(0)=1\) and \(h_6(y)\to\infty\) as \(y\to\infty\), while \(h_6(y^*)<1\), the minimum is attained at \(y^*\).

In \(G_L\), a pendant vertex at \(u\) is dominated only by itself or by \(u\); if it lies in a paired dominating set, its partner is its only neighbor \(u\). So every paired dominating set contains all centers, and the centers form one. Hence \(S^*\) is the only one of size at most \(2L\), and a trial for \(k=2L\) accepts only if its sample lies in \(S^*\). With the anchors at the centers, Lemma D.11.18 holds with equality, and \(W_{S^*}(y)=(y^2+\frac y2+\frac1{64})^L\), so \((1+y)^n/W_{S^*}(y)=2^{n\,h_6(y)}\ge2^{n\,\psi_{\mathrm{pr}}}\) for all \(y\ge0\), with \(n=6L\). By the second inequality of Lemma 3.9, \(\max_t\mu_t(S^*)/\binom nt\le(n+1)2^{-\psi_{\mathrm{pr}}n}\), and by Proposition 3.10 the expected number of trials that run the extension on a subset of \(S^*\), and so the probability of acceptance, is at most \(n^{O(1)}2^{(\psi'-\psi_{\mathrm{pr}})n}\to0\).  ∎

D.11.8 Vertex weights for Directed Steiner Tree and Paired Domination

The two weighted results stated in Sections 12.5 and 12.6 follow from the unweighted proofs through the budget loop of Lemma D.11.6, with the notation of those subsections.

The proof of Theorem 12.8 uses only two properties of \(S\), that \(S\) is rooted and that \(A\) has no leaf in \(S\), and not that \(|S|\) is least.

Corollary D.11.20 (Vertex-weighted Directed Steiner Tree).  For non-terminal weights \(w\colon U\to\{0,1,\dots,w_{\max}\}\), written in unary, the least weight of a Steiner set can be computed by a randomized algorithm with one-sided error in time \(2^{0.457n}(n+w_{\max})^{O(1)}\) and polynomial space.

Proof. We apply Lemma D.11.6 to \(\Phi_{\mathrm{DST}}\) with the weights \(w\) on its universe \(U\); so \(\Phi_\varpi\) has as solutions the rooted Steiner sets of weight at most \(\varpi\), and its least solution weight is the least weight of a Steiner set, because the vertices of a Steiner set that are reachable from \(r\) form a rooted Steiner set of no larger weight. The extension algorithm \(\mathcal{A}_w\) computes the least weight \(\operatorname{OPT}\) of a completion of \(X\) by Lemma 12.7 and accepts if and only if \(w(X)+\operatorname{OPT}\le \varpi\); it ignores \(k\). A witness \(Y\) in (R1) gives \(w(X)+\operatorname{OPT}\le w(X\cup Y)\le\varpi\), and an acceptance yields a rooted Steiner set of weight at most \(\varpi\); so \(\mathcal{A}_w\) satisfies (R1) and (R2) of Lemma D.11.5, and \(g\) is a running-time exponent for it, with \(N^{O(1)}=(n+w_{\max})^{O(1)}\). Let \(\mathcal{B}_\varpi(I,k)\) be Algorithm MLS-\(\theta\) for \(\Phi_\varpi\) with \(\mathcal{A}_w\) and \(g\), with the parameters of the proof of Theorem 12.8. By Lemma D.11.5 with Theorem 3.5, it runs in time \(2^{0.457n}(n+w_{\max})^{O(1)}\) and polynomial space, and it answers no if \(\Phi_\varpi\) has no solution, which is property (B1). For property (B2), let \(S\) be a Steiner set of least weight \(\varpi^*\) that is inclusion-minimal among those; it is rooted, and \(A\) has no leaf in \(S\), since deleting such a leaf would keep a Steiner set of no larger weight. The proof of Theorem 12.8 uses only these two properties of \(S\) to verify condition (\(\star\)) at \(k=|S|\), so \(\mathcal{B}_{\varpi^*}(I,|S|)\) answers yes with probability at least \(1-e^{-\kappa}\). Now Lemma D.11.6 gives the claim, with \(w(U)+1\le n_Uw_{\max}+1\) budgets.  ∎

For unit weights this is Theorem 12.8. We know of no algorithm for vertex-weighted Directed Steiner Tree, or for its undirected special case, with running time \(O(c^n)\) for a constant \(c<\sqrt2\); for undirected Vertex-Weighted Steiner Tree, Buchanan, Wang and Butenko (Buchanan, Wang, and Butenko 2018) give \(O(1.5296^n)\).

Let \(w\colon V\to\{0,\dots,w_{\max}\}\) be vertex weights, and let \(\gamma^w_{\mathrm{pr}}(G)\) be the least weight of a paired dominating set.

Theorem D.11.21 (Weighted Paired Domination).  For graphs \(G\) with \(n\) vertices and no isolated vertex and weights \(w\colon V\to\{0,\dots,w_{\max}\}\) in unary, \(\gamma^w_{\mathrm{pr}}(G)\) can be computed by a randomized algorithm with one-sided error in time \(O(1.6302^n)(n+w_{\max})^{O(1)}\) and polynomial space.

Proof. We apply Lemma D.11.6 to \(\Phi_{\mathrm{PD}}\) with the weights \(w\), so that the solutions of \(\Phi_\varpi\) are the paired dominating sets of weight at most \(\varpi\); Lemma 12.9 with the weights \(w\) and the budget \(\varpi\) is a one-sided-error extension algorithm for \(\Phi_\varpi\) with running-time exponent \(|\operatorname{und}(X)|\), as in Corollary 3.15. Let \(\mathcal{B}_\varpi(G,k)\) be Algorithm MLS-\(\theta\) for \(\Phi_\varpi\) with this extension algorithm and the parameters of the proof of Theorem 12.10, for even \(k\) (and the answer no for odd \(k\), as a paired dominating set has even size). By Theorem 3.5 with Corollary 3.15, it runs in time \(2^{\psi n}(n+w_{\max})^{O(1)}=O(1.6302^n)(n+w_{\max})^{O(1)}\), with \(\psi=0.70502\) as in that proof, and in polynomial space. It answers no if \(\Phi_\varpi\) has no solution, which is property (B1). For property (B2), let \(S\) be a paired dominating set of least weight; Lemma D.11.19 holds for every paired dominating set, so \(S\) satisfies condition (\(\star\)) at \(k=|S|\) and \(\mathcal{B}_{w(S)}(G,|S|)\) answers yes with probability at least \(1-e^{-\kappa}\). Now Lemma D.11.6 gives the claim, with \(w(V)+1\le nw_{\max}+1\) budgets.  ∎

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