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.ESA.2016.58
URN: urn:nbn:de:0030-drops-64001
URL: https://drops.dagstuhl.de/opus/volltexte/2016/6400/
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Kowalik, Lukasz ; Lauri, Juho ; Socala, Arkadiusz

On the Fine-Grained Complexity of Rainbow Coloring

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LIPIcs-ESA-2016-58.pdf (0.7 MB)

Abstract

The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in k colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k >= 2, there is no algorithm for Rainbow k-Coloring running in time 2^{o(n^{3/2})}, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In the Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set S of pairs of vertices and we ask if there is a coloring in which all the pairs in S are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by |S|. We also study Subset Rainbow k-Coloring problem, where we are additionally given an integer q and we ask if there is a coloring in which at least q anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by q and has a kernel of size O(q) for every k >= 2, extending the result of Ananth et al. [FSTTCS 2011]. We believe that our techniques used for the lower bounds may shed some light on the complexity of the classical Edge Coloring problem, where it is a major open question if a 2^{O(n)}-time algorithm exists.

BibTeX - Entry

@InProceedings{kowalik_et_al:LIPIcs:2016:6400,
  author =	{Lukasz Kowalik and Juho Lauri and Arkadiusz Socala},
  title =	{{On the Fine-Grained Complexity of Rainbow Coloring}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{58:1--58:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Piotr Sankowski and Christos Zaroliagis},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6400},
  URN =		{urn:nbn:de:0030-drops-64001},
  doi =		{10.4230/LIPIcs.ESA.2016.58},
  annote =	{Keywords: graph coloring, computational complexity, lower bounds, exponential time hypothesis, FPT algorithms}
}

Keywords: graph coloring, computational complexity, lower bounds, exponential time hypothesis, FPT algorithms
Collection: 24th Annual European Symposium on Algorithms (ESA 2016)
Issue Date: 2016
Date of publication: 18.08.2016

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