For the vertex selection problem $(\sigma,\rho)$-DomSet one is given two fixed sets $\sigma$ and $\rho$ of integers and the task is to decide whether we can select vertices of the input graph, such that, for every selected vertex, the number of selected neighbors is in $\sigma$ and, for every unselected vertex, the number of selected neighbors is in $\rho$. This framework covers Independent Set and Dominating Set for example. We investigate the case when $\sigma$ and $\rho$ are periodic sets with the same period $m\ge 2$, that is, the sets are two (potentially different) residue classes modulo $m$. We study the problem parameterized by treewidth and present an algorithm that solves in time $m^{tw} \cdot n^{O(1)}$ the decision, minimization and maximization version of the problem. This significantly improves upon the known algorithms where for the case $m \ge 3$ not even an explicit running time is known. We complement our algorithm by providing matching lower bounds which state that there is no $(m-\epsilon)^{pw} \cdot n^{O(1)}$ unless SETH fails. For $m = 2$, we extend these bound to the minimization version as the decision version is efficiently solvable.

翻译：对于顶点选择问题$(\sigma,\rho)$-DomSet，给定两个固定的整数集合$\sigma$和$\rho$，任务是判断是否可以从输入图中选择顶点，使得每个被选顶点的已选邻居数属于$\sigma$，而每个未选顶点的已选邻居数属于$\rho$。该框架涵盖例如独立集和支配集问题。我们研究了$\sigma$和$\rho$为具有相同周期$m\ge 2$的周期集的情形，即这两个集合是模$m$的两个（可能不同的）剩余类。我们以树宽为参数研究该问题，并提出一种算法，能在$m^{tw} \cdot n^{O(1)}$时间内解决该问题的判定、最小化和最大化版本。这显著改进了已知算法——对于$m\ge 3$的情况，此前甚至没有显式的运行时间界限。我们通过匹配下界来补充该算法：除非SETH失效，否则不存在$(m-\epsilon)^{pw} \cdot n^{O(1)}$时间算法。对于$m=2$，我们将这些下界扩展到最小化版本，因为其判定版本是高效可解的。