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DOI: 10.4230/LIPIcs.STACS.2011.165
URN: urn:nbn:de:0030-drops-30082
URL: http://drops.dagstuhl.de/opus/volltexte/2011/3008/
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Bodlaender, Hans L. ; Jansen, Bart M. P. ; Kratsch, Stefan

Cross-Composition: A New Technique for Kernelization Lower Bounds

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Abstract

We introduce a new technique for proving kernelization lower bounds, called cross-composition. A classical problem L cross-composes into a parameterized problem $Q$ if an instance of Q with polynomially bounded parameter value can express the logical OR of a sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam (STOC 2008) we show that if an NP-hard problem cross-composes into a parameterized problem Q then Q does not admit a polynomial kernel unless the polynomial hierarchy collapses. Our technique generalizes and strengthens the recent techniques of using OR-composition algorithms and of transferring the lower bounds via polynomial parameter transformations. We show its applicability by proving kernelization lower bounds for a number of important graphs problems with structural (non-standard) parameterizations, e.g., Chromatic Number, Clique, and Weighted Feedback Vertex Set do not admit polynomial kernels with respect to the vertex cover number of the input graphs unless the polynomial hierarchy collapses, contrasting the fact that these problems are trivially fixed-parameter tractable for this parameter. We have similar lower bounds for Feedback Vertex Set.

BibTeX - Entry

@InProceedings{bodlaender_et_al:LIPIcs:2011:3008,
  author =	{Hans L. Bodlaender and Bart M. P. Jansen and Stefan Kratsch},
  title =	{{Cross-Composition: A New Technique for Kernelization Lower Bounds}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
  pages =	{165--176},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Thomas Schwentick and Christoph D{\"u}rr},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2011/3008},
  URN =		{urn:nbn:de:0030-drops-30082},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2011.165},
  annote =	{Keywords: kernelization, lower bounds, parameterized complexity}
}

Keywords: kernelization, lower bounds, parameterized complexity
Seminar: 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)
Issue Date: 2011
Date of publication: 11.03.2011


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