Search-Space Reduction via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion

Authors Bart M. P. Jansen , Ruben F. A. Verhaegh



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Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Ruben F. A. Verhaegh
  • Eindhoven University of Technology, The Netherlands

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Bart M. P. Jansen and Ruben F. A. Verhaegh. Search-Space Reduction via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.28

Abstract

For an optimization problem Π on graphs whose solutions are vertex sets, a vertex v is called c-essential for Π if all solutions of size at most c ⋅ opt contain v. Recent work showed that polynomial-time algorithms to detect c-essential vertices can be used to reduce the search space of fixed-parameter tractable algorithms solving such problems parameterized by the size k of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For example, we give a polynomial-time algorithm for 3-Essential detection for Vertex Multicut, which translates into an algorithm that finds a minimum multicut of an undirected n-vertex graph G in time 2^𝒪(𝓁³)⋅n^𝒪(1), where 𝓁 is the number of vertices in an optimal solution that are not 3-essential. Our positive results are obtained by analyzing the integrality gaps of certain linear programs. Our lower bounds show that for sufficiently small values of c, the detection task becomes NP-hard assuming the Unique Games Conjecture. For example, we show that (2-ε)-Essential detection for Directed Feedback Vertex Set is NP-hard under this conjecture, thereby proving that the existing algorithm that detects 2-essential vertices is best-possible.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Linear programming
  • Theory of computation → Rounding techniques
  • Theory of computation → Fixed parameter tractability
Keywords
  • fixed-parameter tractability
  • essential vertices
  • integrality gap

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