Parameterized Algorithms and Kernels for Rainbow Matching

Abstract

In this paper, we study the NP-complete colorful variant of the classical Matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, this problem asks whether there exists a matching of size at least k such that all the edges in the matching have distinct colors. We first develop a deterministic algorithm that solves Rainbow Matching on paths in time O*(((1+\sqrt{5})/2)^k) and polynomial space. This algorithm is based on a curious combination of the method of bounded search trees and a divide-and-conquer-like approach, where the branching process is guided by the maintenance of an auxiliary bipartite graph where one side captures divided-and-conquered pieces of the path. Our second result is a randomized algorithm that solves Rainbow Matching on general graphs in time O*(2^k) and polynomial space. Here, we show how a result by Bjorklund et al. [JCSS, 2017] can be invoked as a black box, wrapped by a probability-based analysis tailored to our problem. We also complement our two main results by designing kernels for Rainbow Matching on general and bounded-degree graphs.