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Residue Domination in Bounded-Treewidth Graphs

Authors Jakob Greilhuber , Philipp Schepper , Philip Wellnitz

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  • 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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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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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.

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

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.

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