For a well-studied family of domination-type problems, in bounded-treewidth graphs, we investigate whether it is possible to find faster algorithms. For sets $\sigma,\rho$ of non-negative integers, a $(\sigma,\rho)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in \sigma$ for every $u\in S$, and $|N(v)\cap S|\in \rho$ for every $v\not\in S$. The problem of finding a $(\sigma,\rho)$-set (of a certain size) unifies common problems like $\text{Independent Set}$, $\text{Dominating Set}$, $\text{Independent Dominating Set}$, and many others. In an accompanying paper, it is proven that, for all pairs of finite or cofinite sets $(\sigma,\rho)$, there is an algorithm that counts $(\sigma,\rho)$-sets in time $(c_{\sigma,\rho})^{\text{tw}}\cdot n^{O(1)}$ (if a tree decomposition of width $\text{tw}$ is given in the input). Here, $c_{\sigma,\rho}$ is a constant with an intricate dependency on $\sigma$ and $\rho$. Despite this intricacy, we show that the algorithms in the accompanying paper are most likely optimal, i.e., for any pair $(\sigma, \rho)$ of finite or cofinite sets where the problem is non-trivial, and any $\varepsilon>0$, a $(c_{\sigma,\rho}-\varepsilon)^{\text{tw}}\cdot n^{O(1)}$-algorithm counting the number of $(\sigma,\rho)$-sets would violate the Counting Strong Exponential-Time Hypothesis ($\#$SETH). For finite sets $\sigma$ and $\rho$, our lower bounds also extend to the decision version, showing that those algorithms are optimal in this setting as well.

翻译：对于一类广泛研究的支配型问题，我们探讨在树宽有界图中能否设计出更快的算法。设 $\sigma,\rho$ 为非负整数集合，图 $G$ 的 $(\sigma,\rho)$-集是顶点子集 $S$，使得对每个 $u\in S$ 有 $|N(u)\cap S|\in \sigma$，且对每个 $v\not\in S$ 有 $|N(v)\cap S|\in \rho$。寻找（特定大小的）$(\sigma,\rho)$-集的问题统一了诸如独立集（Independent Set）、支配集（Dominating Set）、独立支配集（Independent Dominating Set）等常见问题。在伴随论文中已证明，对所有有限或余有限集合对 $(\sigma,\rho)$，存在算法在 $(c_{\sigma,\rho})^{\text{tw}}\cdot n^{O(1)}$ 时间内计数 $(\sigma,\rho)$-集（若输入中给定宽度为 $\text{tw}$ 的树分解）。其中 $c_{\sigma,\rho}$ 是依赖于 $\sigma$ 和 $\rho$ 的复杂常数。尽管复杂度复杂，我们证明伴随论文中的算法极可能最优：对任意非平凡问题对应的有限或余有限集合对 $(\sigma,\rho)$ 及任意 $\varepsilon>0$，若存在计数 $(\sigma,\rho)$-集的 $(c_{\sigma,\rho}-\varepsilon)^{\text{tw}}\cdot n^{O(1)}$ 算法，则将违反计数强指数时间假说（Counting Strong Exponential-Time Hypothesis, $\#$SETH）。对于有限集合 $\sigma$ 和 $\rho$，我们的下界还扩展至判定版本，表明在该设定下这些算法同样最优。