The exact matching problem is a constrained variant of the maximum matching problem: given a graph with each edge having a weight $0$ or $1$ and an integer $k$, the goal is to find a perfect matching of weight exactly $k$. Mulmuley, Vazirani, and Vazirani (1987) proposed a randomized polynomial-time algorithm for this problem, and it is still open whether it can be derandomized. Very recently, El Maalouly, Steiner, and Wulf (2023) showed that for bipartite graphs there exists a deterministic FPT algorithm parameterized by the (bipartite) independence number. In this paper, by extending a part of their work, we propose a deterministic FPT algorithm in general parameterized by the minimum size of an odd cycle transversal in addition to the (bipartite) independence number. We also consider a relaxed problem called the correct parity matching problem, and show that a slight generalization of an equivalent problem is NP-hard.

翻译：精确匹配问题是最大匹配问题的一个受约束变体：给定一个每条边具有权重$0$或$1$的图以及一个整数$k$，目标是找到一个权重恰好为$k$的完美匹配。Mulmuley、Vazirani和Vazirani（1987）提出了该问题的随机多项式时间算法，但其能否去随机化仍是开放问题。最近，El Maalouly、Steiner和Wulf（2023）证明，对于二分图，存在一个以（二分）独立数为参数化的确定性FPT算法。本文通过扩展他们工作的部分内容，提出了一个以最小奇环横贯规模（附加于（二分）独立数）为参数化的通用确定性FPT算法。我们还考虑了一个称为正确奇偶匹配的放宽问题，并证明其等价问题的一个轻微推广是NP难的。