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DOI: 10.4230/LIPIcs.FSTTCS.2013.153
URN: urn:nbn:de:0030-drops-43689
URL: http://drops.dagstuhl.de/opus/volltexte/2013/4368/
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Chandran, L. Sunil ; Rajendraprasad, Deepak

Inapproximability of Rainbow Colouring

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Abstract

A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one path in which no two edges are coloured the same. The minimum number of colours required to rainbow colour G is called its rainbow connection number. Chakraborty, Fischer, Matsliah and Yuster have shown that it is NP-hard to compute the rainbow connection number of graphs [J. Comb. Optim., 2011]. Basavaraju, Chandran, Rajendraprasad and Ramaswamy have reported an (r+3)-factor approximation algorithm to rainbow colour any graph of radius r [Graphs and Combinatorics, 2012]. In this article, we use a result of Guruswami, Håstad and Sudan on the NP-hardness of colouring a 2-colourable 4-uniform hypergraph using constantly many colours [SIAM J. Comput., 2002] to show that for every positive integer k, it is NP-hard to distinguish between graphs with rainbow connection number 2k+2 and 4k+2. This, in turn, implies that there cannot exist a polynomial time algorithm to rainbow colour graphs with less than twice the optimum number of colours, unless P=NP. The authors have earlier shown that the rainbow connection number problem remains NP-hard even when restricted to the class of chordal graphs, though in this case a 4-factor approximation algorithm is available [COCOON, 2012]. In this article, we improve upon the 4-factor approximation algorithm to design a linear-time algorithm that can rainbow colour a chordal graph G using at most 3/2 times the minimum number of colours if G is bridgeless and at most 5/2 times the minimum number of colours otherwise. Finally we show that the rainbow connection number of bridgeless chordal graphs cannot be polynomial-time approximated to a factor less than 5/4, unless P=NP.

BibTeX - Entry

@InProceedings{chandran_et_al:LIPIcs:2013:4368,
  author =	{L. Sunil Chandran and Deepak Rajendraprasad},
  title =	{{Inapproximability of Rainbow Colouring}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)},
  pages =	{153--162},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-64-4},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{24},
  editor =	{Anil Seth and Nisheeth K. Vishnoi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2013/4368},
  URN =		{urn:nbn:de:0030-drops-43689},
  doi =		{10.4230/LIPIcs.FSTTCS.2013.153},
  annote =	{Keywords: rainbow connectivity, rainbow colouring, approximation hardness}
}

Keywords: rainbow connectivity, rainbow colouring, approximation hardness
Seminar: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)
Issue Date: 2013
Date of publication: 09.12.2013


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