eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2009-02-19
243
254
10.4230/LIPIcs.STACS.2009.1811
article
Hardness and Algorithms for Rainbow Connectivity
Chakraborty, Sourav
Fischer, Eldar
Matsliah, Arie
Yuster, Raphael
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol003-stacs2009/LIPIcs.STACS.2009.1811/LIPIcs.STACS.2009.1811.pdf