We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. 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 standard problems such as Independent Set, Dominating Set, Independent Dominating Set, and many others. For all pairs of finite or cofinite sets $(\sigma,\rho)$, we determine (under standard complexity assumptions) the best possible value $c_{\sigma,\rho}$ such that there is an algorithm that counts $(\sigma,\rho)$-sets in time $c_{\sigma,\rho}^{\sf tw}\cdot n^{O(1)}$ (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 $\rho=\{1\}$, 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 $c_{\sigma,\rho}$, an accompanying paper shows that our algorithms are most likely optimal, that is, for any pair $(\sigma, \rho)$ of finite or cofinite sets where the problem is non-trivial, and any $\varepsilon>0$, a $(c_{\sigma,\rho}-\varepsilon)^{\sf tw}\cdot n^{O(1)}$-algorithm counting the number of $(\sigma,\rho)$-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets $\sigma$ and $\rho$, 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.

翻译：我们研究了一类广泛研究的支配型问题在有界树宽图上的高效求解能力。对于非负整数集合 $\sigma,\rho$，图 $G$ 的 $(\sigma,\rho)$-集是顶点集 $S$，满足对每个 $u\in S$ 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）$(\sigma,\rho)$-集的问题统一了独立集、支配集、独立支配集等众多标准问题。对于所有有限或余有限集合对 $(\sigma,\rho)$，我们在标准复杂度假设下确定了最优常数 $c_{\sigma,\rho}$，使得存在算法在 $c_{\sigma,\rho}^{\sf tw}\cdot n^{O(1)}$ 时间内计数 $(\sigma,\rho)$-集（若输入中给定了宽度为 ${\sf tw}$ 的树分解）。例如，对于对应 $\sigma=\{0\}$ 和 $\rho=\{1\}$ 的精确独立支配集问题（也称为完美码），我们将 [van Rooij, 2020] 的 $3^{\sf tw}\cdot n^{O(1)}$ 算法改进为 $2^{\sf tw}\cdot n^{O(1)}$。尽管 $c_{\sigma,\rho}$ 的定义异常精细，配套论文表明我们的算法极可能是最优的：即对任何问题非平凡的有限或余有限集合对 $(\sigma,\rho)$，以及任何 $\varepsilon>0$，存在 $(c_{\sigma,\rho}-\varepsilon)^{\sf tw}\cdot n^{O(1)}$-算法来计数 $(\sigma,\rho)$-集的数量将违反计数强指数时间假设（#SETH）。对于有限集合 $\sigma$ 和 $\rho$，这些下界也适用于判定版本，因此我们的算法在此场景下也是最优的。相比之下，对于许多余有限集合，我们证明利用代表元集技术可以在判定和优化版本上实现进一步显著改进。