Max-Min Greedy Matching

Abstract

We study the problem of max-min greedy matching introduced in (Eden et al. 2022). The max-min greedy matching problem is a variant of the well known online bipartite matching problem introduced in (Karp et al. 1990). Given a bipartite graph $G=(U \cup V; E)$, the first player selects a priority ordering $\pi$ of $V$. After that the second player selects an arrival order $\sigma$ of $U$ depending on the ordering $\pi$ selected by the first player. Once both players did their selections, the vertices of $U$ arrive in the order specified by $\sigma$. Upon arrival, each vertex in $U$ is matched with its highest-priority neighbor that is still unmatched, if one exists. Here, the highest-priority is defined with respect to $\pi$. The first player aims to maximize the size of the resulting matching, while the second player seeks to minimize it.

We investigate the max-min greedy matching problem. We aim to improve the lower bound of the guaranteed matching size achievable by the first player. We present a linear time algorithm that constructs an ordering $\pi$ such that for every $\sigma$ the ratio between the resulting matching size and the maximum matching size is at least $\frac{5}{9}$. Our result improves the previous best known ratio of $\frac{1}{2} + \frac{1}{86}$ from (Eden et al. 2022). The best known upper bound on the ratio is $\frac{2}{3}$ (Cohen-Addad et al. 2016). Thus, our result makes the gap between the lower and upper bounds for the ratio substantially smaller. Our method introduces a novel ``balanced path decomposition'' technique. This decomposition provides crucial structural properties such as existence of large matchings within these paths; and absence of certain edges between the paths. These structural properties allow us to prove the desired lower bound on the ratio.

Our findings suggest that the achieved matching ratio can be further improved and that the balanced path decomposition may have broader applications.