An Approximation Algorithm for the Exact Matching Problem in Bipartite Graphs

TL;DR

This paper presents a deterministic polynomial-time algorithm for bipartite graphs that finds a perfect matching with red edges close to a target number, improving approximation bounds for the Exact Matching problem.

Contribution

It introduces a new approximation algorithm that guarantees a perfect matching with red edges between one-third and the full target number in bipartite graphs.

Findings

Provides a deterministic polynomial-time algorithm for approximate Exact Matching.

Achieves a red edge count within one-third to the target limit.

Improves upon previous approximation bounds for the problem.

Abstract

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 $k/3 \leq k' \leq k$.