Residue Domination in Bounded-Treewidth Graphs

2024-03-12

Symposium on Theoretical Aspects of Computer Science

AI Summary

This paper studies the modulo-$m$ $(sigma, ho)$-dominating set problem on graphs of bounded treewidth: given two residue classes $sigma, ho subseteq mathbb{Z}_m$ with $m geq 2$, decide/minimize/maximize a vertex subset $S$ such that for each vertex $v$, $|N(v) cap S| mod m in sigma$ if $v in S$, and $in ho$ otherwise. We present the first treewidth-optimal dynamic programming algorithm running in time $m^{operatorname{tw}} cdot n^{O(1)}$ for arbitrary $m geq 2$, achieved via modular state compression and local constraint propagation. Moreover, we establish a matching SETH-tight lower bound of $(m-varepsilon)^{operatorname{pw}} cdot n^{O(1)}$ under pathwidth, resolving the long-standing gap where no explicit upper or lower bounds were known for $m geq 3$.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

For the vertex selection problem $(sigma, ho)$-DomSet one is given two fixed sets $sigma$ and $ ho$ 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 $ ho$ [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 $sigma$ and $ ho$ are two (potentially different) residue classes modulo $mge 2$. 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 $(m-epsilon)^{pw} cdot n^{O(1)}$-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.

Problem

• Investigates vertex selection with residue class constraints
• Improves algorithms for bounded-treewidth graphs
• Provides matching lower bounds for pathwidth

Innovation

• Uses residue classes modulo m for vertex selection
• Algorithm runs in m^tw * n^O(1) time
• Provides matching SETH-based lower bounds