On the Exact Matching Problem in Dense Graphs

TL;DR

This paper advances the understanding of the Exact Matching problem in dense graphs by providing deterministic polynomial-time solutions for various graph classes and quasi-polynomial solutions for random graphs, using local search and generalized algorithms.

Contribution

It offers the first deterministic polynomial-time algorithms for Exact Matching in multiple dense graph classes and extends existing results to new graph categories.

Findings

Deterministic polynomial-time algorithms for complete r-partite graphs, unit interval graphs, and others.

Quasi-polynomial time solution for Erdős-Rényi random graphs G(n, 1/2).

Reproves results for graphs with bounded independence number.

Abstract

In the Exact Matching problem, we are given a graph whose edges are colored red or blue and the task is to decide for a given integer k, if there is a perfect matching with exactly k red edges. Since 1987 it is known that the Exact Matching Problem can be solved in randomized polynomial time. Despite numerous efforts, it is still not known today whether a deterministic polynomial-time algorithm exists as well. In this paper, we make substantial progress by solving the problem for a multitude of different classes of dense graphs. We solve the Exact Matching problem in deterministic polynomial time for complete r-partite graphs, for unit interval graphs, for bipartite unit interval graphs, for graphs of bounded neighborhood diversity, for chain graphs, and for graphs without a complete bipartite t-hole. We solve the problem in quasi-polynomial time for Erd\H{o}s-R\'enyi random graphs G(n, 1/2). We also reprove an earlier result for bounded independence number/bipartite independence number. We use two main tools to obtain these results: A local search algorithm as well as a generalization of an earlier result by Karzanov.