Maximum Cardinality Popular Matchings in Strict Two-sided Preference Lists

We consider the problem of computing a maximum cardinality {\em popular} matching in a bipartite graph $G = (\A\cup\B, E)$ where each vertex $u \in \A\cup\B$ ranks its neighbors in a strict order of preference. This is the same as an instance of the {\em stable marriage} problem with incomplete lists. A matching $M^*$ is said to be popular if there is no matching $M$ such that more vertices are better off in $M$ than in $M^*$. \smallskip Popular matchings have been extensively studied in the case of one-sided preference lists, i.e., only vertices of $\A$ have preferences over their neighbors while vertices in $\B$ have no preferences; polynomial time algorithms have been shown here to determine if a given instance admits a popular matching or not and if so, to compute one with maximum cardinality. It has very recently been shown that for two-sided preference lists, the problem of determining if a given instance admits a popular matching or not is NP-complete. However this hardness result assumes that preference lists have {\em ties}. When preference lists are {\em strict}, it is easy to show that popular matchings always exist since stable matchings always exist and they are popular. But the complexity of computing a maximum cardinality popular matching was unknown. In this paper we show an $O(mn)$ algorithm for this problem, where $n = |\A| + |\B|$ and $m = |E|$