PREPRINT

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs

Jacob Focke, Dániel Marx, Fionn Mc Inerney, Daniel Neuen, Govind S. Sankar, Philipp Schepper, Philip Wellnitz

Submitted on 8 November 2022

Abstract

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets σ,ρ of non-negative integers, a (σ,ρ)-set of a graph G is a set S of vertices such that |N(u)S|σ for every uS, and |N(v)S|ρ for every vS. The problem of finding a (σ,ρ)-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 (σ,ρ), we determine (under standard complexity assumptions) the best possible value cσ,ρ such that there is an algorithm that counts (σ,ρ)-sets in time cσ,ρtwnO(1) (if a tree decomposition of width tw is given in the input). For example, for the Exact Independent Dominating Set problem (also known as Perfect Code) corresponding to σ={0} and ρ={1}, we improve the 3twnO(1) algorithm of [van Rooij, 2020] to 2twnO(1). Despite the unusually delicate definition of cσ,ρ, we show that our algorithms are most likely optimal, i.e., for any pair (σ,ρ) of finite or cofinite sets where the problem is non-trivial, and any ε>0, a (cσ,ρε)twnO(1)-algorithm counting the number of (σ,ρ)-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets σ and ρ, our lower bounds also extend to the decision version, showing that 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.

Preprint

Subjects: Computer Science - Computational Complexity; Computer Science - Data Structures and Algorithms

URL: http://arxiv.org/abs/2211.04278