🤖 AI Summary
This paper addresses the exact counting problem for generalized dominating sets—specifically, (σ,ρ)-sets—on graphs of bounded treewidth, unifying classical problems including independent sets, dominating sets, and independent dominating sets. For all finite or co-finite pairs (σ,ρ), we establish the optimal exponential base $c_{sigma,
ho}$, yielding a tight time complexity bound of $c_{sigma,
ho}^{mathrm{tw}} cdot n^{O(1)}$. Our approach integrates tree-decomposition-based dynamic programming with algebraic acceleration, representative set techniques, and fine-grained lower-bound analysis under the counting Strong Exponential Time Hypothesis (#SETH). The results significantly improve several prior algorithms—for instance, reducing independent dominating set counting to $2^{mathrm{tw}} cdot n^{O(1)}$—and prove that these bounds are #SETH-optimal. This work delivers, for the first time, matching upper and lower bounds for both decision and counting versions of (σ,ρ)-set problems over all finite or co-finite (σ,ρ), thereby achieving theoretical optimality.
📝 Abstract
We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets $sigma,
ho$ of non-negative integers, a $(sigma,
ho)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)cap S|in sigma$ for every $uin S$, and $|N(v)cap S|in
ho$ for every $v
otin S$. The problem of finding a $(sigma,
ho)$-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 $(sigma,
ho)$, we determine (under standard complexity assumptions) the best possible value $c_{sigma,
ho}$ such that there is an algorithm that counts $(sigma,
ho)$-sets in time $c_{sigma,
ho}^{sf tw}cdot n^{O(1)}$ (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 $sigma={0}$ and $
ho={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 $c_{sigma,
ho}$, an accompanying paper shows that our algorithms are most likely optimal, that is, for any pair $(sigma,
ho)$ of finite or cofinite sets where the problem is non-trivial, and any $varepsilon>0$, a $(c_{sigma,
ho}-varepsilon)^{sf tw}cdot n^{O(1)}$-algorithm counting the number of $(sigma,
ho)$-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets $sigma$ and $
ho$, 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.
