Hardness and Algorithms for Rainbow Connectivity

Sourav Chakraborty; Eldar Fischer; Arie Matsliah; Raphael Yuster

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arXiv · 0902.1255

Hardness and Algorithms for Rainbow Connectivity

Sourav Chakraborty·Eldar Fischer·Arie Matsliah·Raphael Yuster

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Abstract

An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The 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 $ε$ > 0, a connected graph with minimum degree at least $εn$ has bounded rainbow connectivity, where the bound depends only on $ε$, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also pre sented.

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Sourav Chakraborty, Eldar Fischer, Arie Matsliah, Raphael Yuster. 2009-02-17. Hardness and Algorithms for Rainbow Connectivity. https://arxiv.org/abs/0902.1255

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