Hardness and Parameterized Algorithms on Rainbow Connectivity problem

TL;DR

This paper investigates the computational complexity of rainbow connectivity problems in graphs, establishing NP-completeness results for fixed parameters, and explores fixed parameter tractability for certain cases.

Contribution

It proves NP-completeness for fixed k in rainbow connectivity problems on bipartite and directed graphs, and shows fixed parameter tractability for maximizing rainbow-connected pairs with two colors.

Findings

NP-Completeness for fixed k >= 3 in bipartite graphs

NP-Completeness for odd k >= 3 in general graphs

Fixed parameter tractability for maximizing rainbow-connected pairs with two colors

Abstract

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colors required to edge color the graph such that G is (strongly) rainbow connected. In this paper we study the rainbow connectivity problem and the strong rainbow connectivity problem from a computational point of view. Our main results can be summarised as below: 1) For every fixed k >= 3, it is NP-Complete to decide whether src(G) <= k even when the graph G is bipartite. 2) For every fixed odd k >= 3, it is NP-Complete to decide whether rc(G) <= k. This resolves one of the open problems posed by Chakraborty et al. (J. Comb. Opt., 2011) where they prove the hardness for the even case. 3) The following problem is fixed parameter tractable: Given a graph G, determine the maximum number of pairs of vertices that can be rainbow connected using two colors. 4) For a directed graph G, it is NP-Complete to decide whether rc(G) <= 2.