On the Exact Matching Problem in Dense Graphs

Nicolas El Maalouly; Sebastian Haslebacher; Lasse Wulf

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arXiv · 2401.03924

On the Exact Matching Problem in Dense Graphs

Nicolas El Maalouly·Sebastian Haslebacher·Lasse Wulf

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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.

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BibTeX RIS

Nicolas El Maalouly, Sebastian Haslebacher, Lasse Wulf. 2024-01-08. On the Exact Matching Problem in Dense Graphs. https://arxiv.org/abs/2401.03924

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