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:
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Edge Clique Cover , parameterized by the number of cliques,
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Directed Edge/Vertex Multiway Cut , parameterized by the size of the cutset, even in the case of two terminals,
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Edge/Vertex Multicut , parameterized by the size of the cutset,
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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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Cygan, M., Kratsch, S., Pilipczuk, M., Pilipczuk, M., Wahlström, M. (2012). Clique Cover and Graph Separation: New Incompressibility Results. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_22
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DOI: https://doi.org/10.1007/978-3-642-31594-7_22
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