On the fine-grained complexity of rainbow coloring

TL;DR

This paper establishes tight complexity bounds for rainbow coloring problems, showing no efficient algorithms under ETH and providing fixed-parameter tractability results for certain variants.

Contribution

It proves ETH-based lower bounds for Rainbow k-Coloring and demonstrates FPT algorithms and kernelization for Subset Rainbow k-Coloring and Maximum Rainbow k-Coloring.

Findings

No $2^{o(n^{3/2})}$ algorithm for Rainbow k-Coloring unless ETH fails.

Subset Rainbow k-Coloring is FPT when parameterized by $|S|$.

Maximum Rainbow k-Coloring is FPT with a linear kernel for parameter $q$.

Abstract

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 $k\ge 2$, there is no algorithm for Rainbow k-Coloring running in time $2^{o(n^{3/2})}$, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], 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 FPT when parameterized by $|S|$. We also study 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 $k\ge 2$ (thus proving that the problem is FPT), extending the result of Ananth et al. [FSTTCS 2011].