Popular Matchings in the Stable Marriage Problem

The input is a bipartite graph G = (A ∪B ,E) where each vertex u ∈ A ∪B ranks its neighbors in a strict order of preference. This is the same as an instance of the 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∗. Any stable matching of G is popular, however such a matching is a minimum cardinality popular matching. We consider the problem of computing a maximum cardinality popular matching in G. It has very recently been shown that when preference lists have ties, the problem of determining if a given instance admits a popular matching or not is NP-complete. When preference lists are strict, popular matchings always exist, however the complexity of computing a maximum cardinality popular matching was unknown. In this paper we give a simple characterization of popular matchings when preference lists are strict and a sufficient condition for a maximum cardinality popular matching. We then show an O(mn0) algorithm for computing a maximum cardinality popular matching, where m = |E| and n0 = min(|A |, |B |).