On the fine-grained complexity of rainbow coloring

The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in k colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k ≥ 2, there is no algorithm for Rainbow k-Coloring running in time 2^o(^{n 3}/²⁾, unless the exponential time hypothesis fails. Motivated by this negative result we consider two parameterized variants of the problem. In the Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [J. Comb. Optim., 21 (2009), pp. 330-347], we are additionally given a set S of pairs of vertices and we ask if there is a coloring in which all the pairs in S are connected by rainbow paths. We show that Subset Rainbow k-Coloring is fixed parameter tractable (FPT) when parameterized by |S|. We also study the Maximum Rainbow k-Coloring problem, where we are additionally given an integer q, and we ask if there is a coloring in which at least q anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by q and has a kernel of size O(q) for every k ≥ 2, extending the result of Ananth, Nasre, and Sarpatwar, in FSTTCS, LIPIcs, Schloss Dagstuhl-Leibniz-Zentum für Informatik, Dagstuhl, Germany, 2011, pp. 241-251. We believe that our techniques used for the lower bounds may shed some light on the complexity of the classical Edge Coloring problem, where it is a major open question if a 2^O(n⁾-time algorithm exists.