In 1982 Papadimitriou and Yannakakis introduced the Exact Matching problem, in which given a red and blue edge-colored graph $G$ and an integer $k$ one has to decide whether there exists a perfect matching in $G$ with exactly $k$ red edges. Even though a randomized polynomial-time algorithm for this problem was quickly found a few years later, it is still unknown today whether a deterministic polynomial-time algorithm exists. This makes the Exact Matching problem an important candidate to test the RP=P hypothesis. In this paper we focus on approximating Exact Matching. While there exists a simple algorithm that computes in deterministic polynomial-time an almost perfect matching with exactly $k$ red edges, not a lot of work focuses on computing perfect matchings with almost $k$ red edges. In fact such an algorithm for bipartite graphs running in deterministic polynomial-time was published only recently (STACS'23). It outputs a perfect matching with $k'$ red edges with the guarantee that $0.5k \leq k' \leq 1.5k$. In the present paper we aim at approximating the number of red edges without exceeding the limit of $k$ red edges. We construct a deterministic polynomial-time algorithm, which on bipartite graphs computes a perfect matching with $k'$ red edges such that $k/3 \leq k' \leq k$.

翻译：1982年Papadimitriou和Yannakakis提出了精确匹配问题：给定一个红蓝边染色的图$G$和一个整数$k$，需要判断$G$中是否存在恰好包含$k$条红边的完美匹配。虽然该问题在数年后很快找到了随机多项式时间算法，但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RP=P猜想的重要候选案例。本文聚焦于精确匹配的近似求解。尽管存在简单算法可在确定性多项式时间内计算含恰好$k$条红边的近完美匹配，但针对计算含接近$k$条红边的完美匹配的研究仍较少。事实上，针对二分图且运行于确定性多项式时间的此类算法直到近期才被提出（STACS'23）。该算法输出含$k'$条红边的完美匹配，满足$0.5k \leq k' \leq 1.5k$。本文旨在近似红边数量且不超过$k$的上限。我们构造了一个确定性多项式时间算法，该算法针对二分图计算含$k'$条红边的完美匹配，使得$k/3 \leq k' \leq k$。