Clique Cover and Graph Separation: New Incompressibility Results

  • Marek Cygan
  • Stefan Kratsch
  • Marcin Pilipczuk
  • Michał Pilipczuk
  • Magnus Wahlström
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

The field of kernelization studies polynomial-time preprocessing routines for hard problems in the framework of parameterized complexity. In this paper we show that, unless \(\textrm{NP} \subseteq \textrm{coNP}/\textrm{poly}\) and the polynomial hierarchy collapses up to its third level, the following parameterized problems do not admit a polynomial-time preprocessing algorithm that reduces the size of an instance to polynomial in the parameter:
  • Edge Clique Cover , parameterized by the number of cliques,

  • Directed Edge/VertexMultiway Cut , parameterized by the size of the cutset, even in the case of two terminals,

  • Edge/VertexMulticut , parameterized by the size of the cutset,

  • and k-Way Cut , parameterized by the size of the cutset.

The existence of a polynomial kernelization for Edge Clique Cover was a seasoned veteran in open problem sessions. Furthermore, our results complement very recent developments in designing parameterized algorithms for cut problems by Marx and Razgon [STOC’11], Bousquet et al. [STOC’11], Kawarabayashi and Thorup [FOCS’11] and Chitnis et al. [SODA’12].

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Marek Cygan
    • 1
  • Stefan Kratsch
    • 2
  • Marcin Pilipczuk
    • 3
  • Michał Pilipczuk
    • 4
  • Magnus Wahlström
    • 5
  1. 1.IDSIAUniversity of LuganoSwitzerland
  2. 2.Utrecht UniversityUtrechtThe Netherlands
  3. 3.Institute of InformaticsUniversity of WarsawPoland
  4. 4.Department of InformaticsUniversity of BergenNorway
  5. 5.Max-Planck-Institute for InformaticsSaarbrückenGermany

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