Hardness and Algorithms for Rainbow Connectivity
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.
243-254
Regular Paper
Sourav
Chakraborty
Sourav Chakraborty
Eldar
Fischer
Eldar Fischer
Arie
Matsliah
Arie Matsliah
Raphael
Yuster
Raphael Yuster
10.4230/LIPIcs.STACS.2009.1811
Creative Commons Attribution-NoDerivs 3.0 Unported license
https://creativecommons.org/licenses/by-nd/3.0/legalcode