A Size-Popularity Tradeoff in the Stable Marriage Problem

Abstract

Given a bipartite graph $G = (\mathcal{A}\cup\mathcal{B}, E)$ where each vertex ranks its neighbors in a strict order of preference, the problem of computing a stable matching is classical and well studied. A stable matching has size at least $\frac{1}{2}|M_{\max}|$, where $M_{\max}$ is a maximum size matching in $G$, and there are simple examples where this bound is tight. It is known that a stable matching is a minimum size popular matching. A matching $M$ is said to be popular if there is no matching where more vertices are better off than in $M$. In this paper we show the first linear time algorithm for computing a maximum size popular matching in $G$. A maximum size popular matching is guaranteed to have size at least $\frac{2}{3}|M_{\max}|$, and this bound is tight. We then consider the following problem: is there a maximum size matching $M^*$ that is popular within the set of maximum size matchings in $G$, that is, $|M^*| = |M_{\max}|$ and there is no maximum size matching that is more popular than $M^*$? We show that such a matching $M^*$ always exists and can be computed in $O(mn_0)$ time, where $m = |E|$ and $n_0 = \min(|\mathcal{A}|,|\mathcal{B}|)$. Though the above matching $M^*$ is popular restricted to the set of maximum size matchings, in the entire set of matchings in $G$, its unpopularity factor could be as high as $n_0-1$. On the other hand, a maximum size popular matching could be of size only $\frac{2}{3}|M_{\max}|$. In between these two extremes, we show there is an entire spectrum of matchings: for any integer $k$, where $2 \le k \le n_0$, there is a matching $M_k$ in $G$ of size at least $\frac{k}{k+1}|M_{\max}|$ whose unpopularity factor is at most $k-1$. Also, such a matching $M_k$ can be computed in $O(km)$ time by a simple generalization of our maximum size popular matching algorithm.