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Algorithmica

, Volume 80, Issue 8, pp 2221–2239 | Cite as

Approximability of Clique Transversal in Perfect Graphs

  • Samuel Fiorini
  • R. Krithika
  • N. S. Narayanaswamy
  • Venkatesh Raman
Article
  • 241 Downloads

Abstract

Given an undirected simple graph G, a set of vertices is an r-clique transversal if it has at least one vertex from every r-clique. Such sets generalize vertex covers as a vertex cover is a 2-clique transversal. Perfect graphs are a well-studied class of graphs on which a minimum weight vertex cover can be obtained in polynomial time. Further, an r-clique transversal in a perfect graph is also a set of vertices whose deletion results in an \((r-1)\)-colorable graph. In this work, we study the problem of finding a minimum weight r-clique transversal in a perfect graph. This problem is known to be \(\mathsf {NP}\)-hard for \(r \ge 3\) and admits a straightforward r-approximation algorithm. We describe two different \(\frac{r+1}{2}\)-approximation algorithms for the problem. Both the algorithms are based on (different) linear programming relaxations. The first algorithm employs the primal–dual method while the second uses rounding based on a threshold value. We also show that the problem is APX-hard and describe hardness results in the context of parameterized algorithms and kernelization.

Keywords

r-Clique transversal Odd cycle transversal Perfect graphs Approximation algorithms Linear programming 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Samuel Fiorini
    • 1
  • R. Krithika
    • 2
  • N. S. Narayanaswamy
    • 3
  • Venkatesh Raman
    • 2
  1. 1.Département de MathématiqueUniversité libre de BruxellesBrusselsBelgium
  2. 2.The Institute of Mathematical SciencesHBNIChennaiIndia
  3. 3.Department of Computer Science and EngineeringIndian Institute of Technology MadrasChennaiIndia

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