A Size-Popularity Tradeoff in the Stable Marriage Problem

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...