Inapproximability of Rainbow Colouring

A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one path in which no two edges are coloured the same. The minimum number of colours required to rainbow colour G is called its rainbow connection number. Chakraborty, Fischer, Matsliah and Yuster have shown that it is NP-hard to compute the rainbow connection number of graphs [J. Comb. Optim., 2011]. Basavaraju, Chandran, Rajendraprasad and Ramaswamy have reported an (r+3)-factor approximation algorithm to rainbow colour any graph of radius r [Graphs and Combinatorics, 2012]. In this article, we use a result of Guruswami, Hastad and Sudan on the NP-hardness of colouring a 2-colourable 4-uniform hypergraph using constantly many colours [SIAM J. Comput., 2002] to show that for every positive integer k, it is NP-hard to distinguish between graphs with rainbow connection number 2k+2 and 4k+2. This, in turn, implies that there cannot exist a polynomial time algorithm to rainbow colour graphs with less than twice the optimum number of colours, unless P=NP. The authors have earlier shown that the rainbow connection number problem remains NP-hard even when restricted to the class of chordal graphs, though in this case a 4-factor approximation algorithm is available [COCOON, 2012]. In this article, we improve upon the 4-factor approximation algorithm to design a linear-time algorithm that can rainbow colour a chordal graph G using at most 3/2 times the minimum number of colours if G is bridgeless and at most 5/2 times the minimum number of colours otherwise. Finally we show that the rainbow connection number of bridgeless chordal graphs cannot be polynomial-time approximated to a factor less than 5/4, unless P=NP.