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Hardness and Algorithms for Rainbow Connectivity

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Abstract

An edge-colored graph $G$ is {\em rainbow connected} if any two vertices are connected by a path whose edges have distinct colors. The {\em rainbow connectivity} of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In addition to being a natural combinatorial problem, the rainbow connectivity problem is motivated by applications in cellular networks. In this paper we give the first proof that computing $rc(G)$ is NP-Hard. In fact, we prove that it is already NP-Complete to decide if $rc(G)=2$, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every $\epsilon >0$, a connected graph with minimum degree at least $\epsilon n$ has bounded rainbow connectivity, where the bound depends only on $\epsilon$, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.

BibTeX - Entry

@InProceedings{chakraborty_et_al:LIPIcs:2009:1811,
  author =	{Sourav Chakraborty and Eldar Fischer and Arie Matsliah and Raphael Yuster},
  title =	{{Hardness and Algorithms for Rainbow Connectivity}},
  booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{243--254},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-09-5},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{3},
  editor =	{Susanne Albers and Jean-Yves Marion},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/1811},
  URN =		{urn:nbn:de:0030-drops-18115},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2009.1811},
  annote =	{Keywords: }
}

Seminar: 26th International Symposium on Theoretical Aspects of Computer Science
Issue date: 2009
Date of publication: 2009


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