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.

翻译：在精确匹配问题中，给定一个边被染成红色或蓝色的图，任务是对给定的整数k判断是否存在恰好包含k条红边的完美匹配。自1987年以来，已知精确匹配问题可以在随机多项式时间内求解。尽管经过了多次努力，至今仍不清楚是否存在确定性的多项式时间算法。在本文中，我们通过解决多种不同类别稠密图的问题取得了重大进展。我们在确定性多项式时间内解决了完全r部图、单位区间图、二分单位区间图、有界邻域多样性图、链图以及无完全二分t-孔图的精确匹配问题。我们在Erdős–Rényi随机图G(n, 1/2)上以拟多项式时间解决了该问题。我们还重新证明了关于有界独立数/二分独立数的早期结果。我们使用两个主要工具来获得这些结果：一个局部搜索算法以及Karzanov早期结果的推广。