Optimization problems in multiple-interval graphs

Ayelet Butman, Danny Hermelin, Moshe Lewenstein, Dror Rawitz

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multiple-interval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We describe applications for each one of these problems, and then proceed to discuss approximation algorithms for them. Our results can be summarized as follows: Let t be the number of intervals associated with each vertex in a given multiple-interval graph. For Minimum Vertex Cover, we give a (2-1/t)-approximation algorithm which also works when a t-interval representation of our given graph is absent. Following this, we give a t2-approximation algorithm for Minimum Dominating Set which adapts well to more general variants of the problem. We then proceed to prove that Maximum Clique is NP-hard already for 3-interval graphs, and provide a (t 2-t+1)/2-approximation algorithm for general values of t ≥ 2, using bounds proven for the so-called transversal number of t-interval families.

Original languageEnglish
Article number40
JournalACM Transactions on Algorithms
Volume6
Issue number2
DOIs
StatePublished - 1 Mar 2010
Externally publishedYes

Keywords

  • Approximation algorithms
  • Maximum clique
  • Minimum dominating set
  • Minimum vertex cover
  • Multiple-interval graphs
  • T-interval graphs

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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