Hardness and Algorithms for Rainbow Connectivity

Authors Sourav Chakraborty, Eldar Fischer, Arie Matsliah, Raphael Yuster

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Sourav Chakraborty
Eldar Fischer
Arie Matsliah
Raphael Yuster

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Sourav Chakraborty, Eldar Fischer, Arie Matsliah, and Raphael Yuster. Hardness and Algorithms for Rainbow Connectivity. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 243-254, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)


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.


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