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Automata, Languages, and Programming

Volume 7391 of the series Lecture Notes in Computer Science pp 254-265

Clique Cover and Graph Separation: New Incompressibility Results

  • Marek CyganAffiliated withIDSIA, University of Lugano
  • , Stefan KratschAffiliated withUtrecht University
  • , Marcin PilipczukAffiliated withInstitute of Informatics, University of Warsaw
  • , Michał PilipczukAffiliated withDepartment of Informatics, University of Bergen
  • , Magnus WahlströmAffiliated withMax-Planck-Institute for Informatics

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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:
  • Edge Clique Cover , parameterized by the number of cliques,

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

  • Edge/Vertex Multicut , 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].