A Complexity Dichotomy for Finding Disjoint Solutions of Vertex Deletion Problems

  • Authors:

  • Michael R. Fellows;Jiong Guo;Hannes Moser;Rolf Niedermeier

  • Affiliations:

  • PC Research Unit, Office of DVC (Research), University of Newcastle, Callaghan, Australia 2308;Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany D-07743;Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany D-07743;Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany D-07743

  • Venue:
  • MFCS '09 Proceedings of the 34th International Symposium on Mathematical Foundations of Computer Science 2009
  • Year:
  • 2009

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Abstract

We investigate the computational complexity of a general "compression task" centrally occurring in the recently developed technique of iterative compression for exactly solving NP-hard minimization problems. The core issue (particularly but not only motivated by iterative compression) is to determine the computational complexity of, given an already inclusion-minimal solution for an underlying (typically NP-hard) vertex deletion problem in graphs, to find a better disjoint solution. The complexity of this task is so far lacking a systematic study. We consider a large class of vertex deletion problems on undirected graphs and show that, except for few cases which are polynomial-time solvable, the others are NP-complete. This class includes problems such as Vertex Cover (here the corresponding compression task is decidable in polynomial time) or Undirected Feedback Vertex Set (here the corresponding compression task is NP-complete).