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Residue Domination in Bounded-Treewidth Graphs

Authors Jakob Greilhuber , Philipp Schepper , Philip Wellnitz

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Author Details

Jakob Greilhuber
  • TU Wien, Austria
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Philipp Schepper
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Philip Wellnitz
  • National Institute of Informatics, Tokyo, Japan
  • The Graduate University for Advanced Studies, SOKENDAI, Tokyo, Japan

Funding

  • Greilhuber, Jakob: Part of Saarbrücken Graduate School of Computer Science, Germany.
  • Schepper, Philipp: Part of Saarbrücken Graduate School of Computer Science, Germany.

Acknowledgements

The work of Jakob Greilhuber has been carried out mostly during a summer internship at the Max Planck Institute for Informatics, Saarbrücken, Germany. The work of Philip Wellnitz was partially carried out at the Max Planck Institute for Informatics.

Cite As Get BibTex

Jakob Greilhuber, Philipp Schepper, and Philip Wellnitz. Residue Domination in Bounded-Treewidth Graphs. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 41:1-41:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.41

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized Complexity
  • Treewidth
  • Generalized Dominating Set
  • Strong Exponential Time Hypothesis

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Related Versions

  • The Master’s Thesis of Jakob Greilhuber [Jakob Greilhuber, 2024] is based on the lower bound results of this work.
  • Full Version https://arxiv.org/abs/2403.07524

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