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.ICALP.2018.86
URN: urn:nbn:de:0030-drops-90903
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9090/
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Lampis, Michael

Finer Tight Bounds for Coloring on Clique-Width

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LIPIcs-ICALP-2018-86.pdf (0.4 MB)

Abstract

We revisit the complexity of the classical k-Coloring problem parameterized by clique-width. This is a very well-studied problem that becomes highly intractable when the number of colors k is large. However, much less is known on its complexity for small, concrete values of k. In this paper, we completely determine the complexity of k-Coloring parameterized by clique-width for any fixed k, under the SETH. Specifically, we show that for all k >= 3,epsilon>0, k-Coloring cannot be solved in time O^*((2^k-2-epsilon)^{cw}), and give an algorithm running in time O^*((2^k-2)^{cw}). Thus, if the SETH is true, 2^k-2 is the "correct" base of the exponent for every k.
Along the way, we also consider the complexity of k-Coloring parameterized by the related parameter modular treewidth (mtw). In this case we show that the "correct" running time, under the SETH, is O^*({k choose floor[k/2]}^{mtw}). If we base our results on a weaker assumption (the ETH), they imply that k-Coloring cannot be solved in time n^{o(cw)}, even on instances with O(log n) colors.

BibTeX - Entry

@InProceedings{lampis:LIPIcs:2018:9090,
  author =	{Michael Lampis},
  title =	{{Finer Tight Bounds for Coloring on Clique-Width}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{86:1--86:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9090},
  URN =		{urn:nbn:de:0030-drops-90903},
  doi =		{10.4230/LIPIcs.ICALP.2018.86},
  annote =	{Keywords: Clique-width, SETH, Coloring}
}

Keywords: Clique-width, SETH, Coloring
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018

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