License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2017.22
URN: urn:nbn:de:0030-drops-85500
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8550/
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Jansen, Bart M. P. ; Pieterse, Astrid

Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

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LIPIcs-IPEC-2017-22.pdf (0.6 MB)

Abstract

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring.
First, we use a recent technique of finding redundant constraints by representing them as low-degree polynomials, to obtain a kernel of bitsize O(k^(q-1) log k) for q-Coloring parameterized by Vertex Cover for any q >= 3. This size bound is optimal up to k^o(1) factors assuming NP is not a subset of coNP/poly, and improves on the previous-best kernel of size O(k^q). Our second result shows that 3-Coloring does not admit non-trivial sparsification: assuming NP is not a subset of coNP/poly, the parameterization by the number of vertices n admits no (generalized) kernel of size O(n^(2-e)) for any e > 0. Previously, such a lower bound was only known for coloring with q >= 4 colors.

BibTeX - Entry

@InProceedings{jansen_et_al:LIPIcs:2018:8550,
  author =	{Bart M. P. Jansen and Astrid Pieterse},
  title =	{{Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials}},
  booktitle =	{12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
  pages =	{22:1--22:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-051-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{89},
  editor =	{Daniel Lokshtanov and Naomi Nishimura},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8550},
  URN =		{urn:nbn:de:0030-drops-85500},
  doi =		{10.4230/LIPIcs.IPEC.2017.22},
  annote =	{Keywords:  graph coloring, kernelization, sparsification}
}

Keywords: graph coloring, kernelization, sparsification
Collection: 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)
Issue Date: 2018
Date of publication: 02.03.2018

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