Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs Part I: Algorithmic Results

TL;DR

This work analyzes counting $(\sigma,\rho)$-sets on graphs of bounded treewidth, a unifying framework for domination-type problems such as Independent Set, Dominating Set, and Perfect Code. It introduces the notion of ${\mathrm{m}}$-structured sets to obtain improved DP-based counting bounds, and develops both structured-language compression and fast-join techniques (via generalized convolution) to reduce the number of DP states. A second axis uses representative sets to achieve faster decision (and some optimization) algorithms when cofiniteness holds, with running times that depend on the number of missing elements rather than the largest missing one. Together with an accompanying paper establishing tight lower bounds under #SETH, the results identify near-optimal bases $c_{\sigma,\rho}$ for counting $(\sigma,\rho)$-sets in bounded-treewidth graphs, and demonstrate significant improvements for the Exact Independent Dominating Set case (Perfect Code) beyond earlier $3^{\sf tw}$-time procedures. The methods bridge algebraic techniques, structured DP, and representative-set theory to obtain tight, conditionally optimal bounds for a broad family of LC-VSP problems on bounded-treewidth graphs.

Abstract

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets $σ,ρ$ of non-negative integers, a $(σ,ρ)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in σ$ for every $u\in S$, and $|N(v)\cap S|\in ρ$ for every $v\not\in S$. The problem of finding a $(σ,ρ)$-set (of a certain size) unifies standard problems such as Independent Set, Dominating Set, Independent Dominating Set, and many others. For all pairs of finite or cofinite sets $(σ,ρ)$, we determine (under standard complexity assumptions) the best possible value $c_{σ,ρ}$ such that there is an algorithm that counts $(σ,ρ)$-sets in time $c_{σ,ρ}^{\sf tw}\cdot n^{O(1)}$ (if a tree decomposition of width ${\sf tw}$ is given in the input). For example, for the Exact Independent Dominating Set problem (also known as Perfect Code) corresponding to $σ=\{0\}$ and $ρ=\{1\}$, we improve the $3^{\sf tw}\cdot n^{O(1)}$ algorithm of [van Rooij, 2020] to $2^{\sf tw}\cdot n^{O(1)}$. Despite the unusually delicate definition of $c_{σ,ρ}$, an accompanying paper shows that our algorithms are most likely optimal, that is, for any pair $(σ, ρ)$ of finite or cofinite sets where the problem is non-trivial, and any $\varepsilon>0$, a $(c_{σ,ρ}-\varepsilon)^{\sf tw}\cdot n^{O(1)}$-algorithm counting the number of $(σ,ρ)$-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets $σ$ and $ρ$, these lower bounds also extend to the decision version, and hence, our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets.

Figures (4)

• Figure 2.1: For Dominating Set (where $\sigma = \{0, 1, 2, \dots\}, \rho = \{1, 2, \dots\}$), it is easy to construct an example where any string $y \in {\mathbb A}^{X_t} = \{\sigma_0,\rho_0,\rho_1\}^{X_t}$ is compatible with $t$: we depict selected vertices as encircled blue and unselected vertices without a selected neighbor as a hollow black circle. Observe that after selecting $w$, any selection of the remaining vertices constitutes a valid partial solution. Hence, there are $3^{|X_t|} = (s_{\textup{top}} + r_{\textup{top}} + 2)^{|X_t|}$ compatible strings in this case.
• Figure 2.2: An example for partial solutions and edges between them for Perfect Code.
• Figure 3.1: A graph $G$ and subsets of vertices $U$ and $S$. For $\sigma = \{2, 4\}, \rho = \{ 1 \}$, the set $S$ is a partial solution (with respect to $U$), as every blue vertex $s \in S \setminus U$ satisfies $|N(s) \cap S| \in \{2,4\} = \sigma$ and every black vertex $v \in V(G) \setminus (S \cup U)$ satisfies $|N(v) \cap S| \in \{1\} = \rho$. The depicted set $S$ corresponds to the compatible string $\sigma_3\sigma_3\sigma_2\rho_1\rho_0$ (written above $G$). Note that $S$ would not be a partial solution for $\sigma = \{4\}$, as every blue vertex but one has only $2$ neighbors in $S$.
• Figure 4.1: A graph $G$ with portals $U$. For $\rho = \{ 1, 3 \}, \sigma = \{2, 4\}$ (which are $2$-structured) the strings $x = \sigma_3\sigma_3\sigma_2\rho_1\rho_0$ and $y = \rho_2\sigma_1\rho_3\sigma_0\sigma_1$ are compatible with $(G,U)$; the corresponding partial solutions $S_x$ and $S_y$, as well as the partitions of $U$ are depicted above. We have $|S_x\setminus U| = |S_y\setminus U| = 4$ and $\vec{\sigma}(x)\cdot \vec{w}_{{\mathrm{m}}}(y) = (1,1,1,0,0) \cdot (0,1,1,0,1) = 2 \equiv_2 2 = (0,1,0,1,1) \cdot (1,1,0,1,0) = \vec{\sigma}(y)\cdot \vec{w}_{{\mathrm{m}}}(x).$

Theorems & Definitions (75)

• Theorem 1.1: Rooij20
• Definition 1.2
• Theorem 1.3
• Theorem 1.4: FockeMMNSSW23ii
• Theorem 1.5: FockeMMNSSW23ii
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• proof