Clique cover and graph separation: new incompressibility results

  • Authors:

  • Marek Cygan;Stefan Kratsch;Marcin Pilipczuk;Michał Pilipczuk;Magnus Wahlström

  • Affiliations:

  • IDSIA, University of Lugano, Switzerland;Utrecht University, Utrecht, The Netherlands;Institute of Informatics, University of Warsaw, Poland;Department of Informatics, University of Bergen, Norway;Max-Planck-Institute for Informatics, Saarbrücken, Germany

  • Venue:
  • ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
  • Year:
  • 2012

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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: Ѿ 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].