Complexity and approximation of satisfactory partition problems

  • Authors:

  • Cristina Bazgan;Zsolt Tuza;Daniel Vanderpooten

  • Affiliations:

  • LAMSADE, Université Paris-Dauphine, France;Computer and Automation Institute, Hungarian Academy of Sciences, Budapest and Department of Computer Science, University of Veszprém, Hungary;LAMSADE, Université Paris-Dauphine, France

  • Venue:
  • COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
  • Year:
  • 2005

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Abstract

The Satisfactory Partition problem consists of deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler in 1998 and further studied by other authors but its complexity remained open until now. We prove in this paper that Satisfactory Partition, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. We also study approximation results for the latter problem, showing that it has no polynomial-time approximation scheme, whereas a constant approximation can be obtained in polynomial time. Similar results hold for balanced partitions where each vertex is required to have at most as many neighbors in its part as in the other part.