Residue Domination in Bounded-Treewidth Graphs

Summary

For the vertex selection problem (σ,ρ)-DomSet one is given two fixed sets σ and ρ 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 σ and, for every unselected vertex, the number of selected neighbors is in ρ [Telle, Nord. J. Comp. 1994]. This framework covers many fundamental graph problems such as Independent Set and Dominating Set. We significantly extend the recent result by Focke et al. [SODA 2023] to investigate the case when σ and ρ are two (potentially different) residue classes modulo m ≥ 2. We study the problem parameterized by treewidth and present an algorithm that solves in time m^tw ⋅ n^O(1) the decision, minimization and maximization version of the problem. This significantly improves upon the known algorithms where for the case m ≥ 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-ε)^pw ⋅ n^O(1)-time algorithm parameterized by pathwidth pw, unless SETH fails. For m = 2, we extend these bounds to the minimization version as the decision version is efficiently solvable.