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.2018.47
URN: urn:nbn:de:0030-drops-90510
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9051/
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Dvorák, Zdenek ; Kawarabayashi, Ken-ichi

Additive Non-Approximability of Chromatic Number in Proper Minor-Closed Classes

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LIPIcs-ICALP-2018-47.pdf (0.4 MB)

Abstract

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k >= 1, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using at most chi(G)+k-1 colors. More generally, for every k >= 1 and 1 <=beta <=4/3, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using less than beta chi(G)+(4-3 beta)k colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes.
We also give somewhat weaker non-approximability bound for K_{4k+1}-minor-free graphs with no cliques of size 4. On the positive side, we present an additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles.

BibTeX - Entry

@InProceedings{dvork_et_al:LIPIcs:2018:9051,
  author =	{Zdenek Dvor{\'a}k and Ken-ichi Kawarabayashi},
  title =	{{Additive Non-Approximability of Chromatic Number in Proper Minor-Closed Classes}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{47:1--47:12},
  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/2018/9051},
  URN =		{urn:nbn:de:0030-drops-90510},
  doi =		{10.4230/LIPIcs.ICALP.2018.47},
  annote =	{Keywords: non-approximability, chromatic number, minor-closed classes}
}

Keywords: non-approximability, chromatic number, minor-closed classes
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018

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