A Complexity Dichotomy for Generalized Rainbow Matchings Based on Color Classes

TL;DR

This paper establishes a clear complexity dichotomy for the Maximum Rainbow Matching problem, showing it is polynomial-time solvable when most color classes are complete multipartite graphs and NP-hard otherwise.

Contribution

It introduces a complexity classification based on color class structure and develops algorithms and hardness proofs for these cases.

Findings

Polynomial-time algorithm for almost all color classes being complete multipartite graphs.

NP-hardness proven for color classes that are small subgraphs like matchings, paths, or paw.

Introduction of color-closed graph classes for complexity analysis.

Abstract

Given an edge-colored graph, the Maximum Rainbow Matching problem asks for a maximum-cardinality matching of the graph that contains at most one edge from each color. We provide the following complexity dichotomy for this problem based on the structure of the color classes: Maximum Rainbow Matching admits a polynomial-time algorithm if almost every color class is a complete multipartite graph and it is NP-hard otherwise. To prove the NP-hardness-part of the dichotomy, we first show that the problem remains NP-hard even if every color class is a subgraph on four vertices that is either a matching of size two, a path on four vertices or a paw. We then leverage this result to all color classes that are not complete multipartite graphs. For this purpose, we introduce color-closed graph classes, which seem to be an appropriate notion for obtaining complexity classifications for rainbow problems and may be of independent interest. To prove the positive part of the dichotomy, we show that the problem essentially reduces to computing a maximum $(l, u)$-matching, where we heavily exploit that almost all color classes are complete multipartite graphs. In the case where all color classes are complete multipartite, we provide a polynomial-time algorithm that computes a maximum matching containing at most $m_i$ edges from each color class $i$.