Given an integer $k$ and a graph where every edge is colored either red or blue, the goal of the exact matching problem is to find a perfect matching with the property that exactly $k$ of its edges are red. Soon after Papadimitriou and Yannakakis (JACM 1982) introduced the problem, a randomized polynomial-time algorithm solving the problem was described by Mulmuley et al. (Combinatorica 1987). Despite a lot of effort, it is still not known today whether a deterministic polynomial-time algorithm exists. This makes the exact matching problem an important candidate to test the popular conjecture that the complexity classes P and RP are equal. In a recent article (MFCS 2022), progress was made towards this goal by showing that for bipartite graphs of bounded bipartite independence number, a polynomial time algorithm exists. In terms of parameterized complexity, this algorithm was an XP-algorithm parameterized by the bipartite independence number. In this article, we introduce novel algorithmic techniques that allow us to obtain an FPT-algorithm. If the input is a general graph we show that one can at least compute a perfect matching $M$ which has the correct number of red edges modulo 2, in polynomial time. This is motivated by our last result, in which we prove that an FPT algorithm for general graphs, parameterized by the independence number, reduces to the problem of finding in polynomial time a perfect matching $M$ with at most $k$ red edges and the correct number of red edges modulo 2.

翻译：给定整数 $k$ 和一个每条边染有红色或蓝色的图，精确匹配问题的目标是找到一个具有恰好 $k$ 条红边的完美匹配。在 Papadimitriou 和 Yannakakis (JACM 1982) 提出该问题后不久，Mulmuley 等人 (Combinatorica 1987) 描述了一种随机多项式时间算法。尽管经过大量努力，至今仍未知是否存在确定性多项式时间算法。这使得精确匹配问题成为检验复杂度类 P 与 RP 相等这一流行猜想的重要候选对象。在近期一篇论文 (MFCS 2022) 中，通过证明对于具有有界二部独立数的二部图存在多项式时间算法，朝着该目标取得了进展。从参数化复杂度角度看，该算法是以二部独立数为参数的 XP 算法。在本文中，我们引入了新颖的算法技术，从而能够获得一个 FPT 算法。若输入为一般图，我们证明至少可以在多项式时间内计算出一个红边数量模 2 正确的完美匹配 $M$。这一结论源于我们的最后一项结果：对于一般图，以独立数为参数的 FPT 算法可归约为在多项式时间内寻找一个红边数不超过 $k$ 且红边数量模 2 正确的完美匹配 $M$ 的问题。