Popular matchings in the stable marriage problem

We consider the problem of computing a maximum cardinality popular matching in a bipartite graph G=(A∪B,E)G=(A∪B,E) where each vertex u∈A∪Bu∈A∪B ranks its neighbors in a strict order of preference. Such a graph is called an instance of the stable marriage problem with strict preferences and incomplete lists. A matching M⁎M⁎ is popular if for every matching M in G, the number of vertices that prefer M   to M⁎M⁎ is at most the number of vertices that prefer M⁎M⁎ to M. Every stable matching of G is popular, however a stable matching is a minimum cardinality popular matching. The complexity of computing a maximum cardinality popular matching was unknown.In this paper we show a simple characterization of popular matchings in G=(A∪B,E)G=(A∪B,E). We also show a sufficient condition for a popular matching to be a maximum cardinality popular matching. We construct a matching that satisfies our characterization and sufficient condition in O(mn0)O(mn0) time, where m=|E|m=|E| and n0=min(|A|,|B|)n0=min(|A|,|B|). Thus the maximum cardinality popular matching problem in G=(A∪B,E)G=(A∪B,E) can be solved in O(mn0)O(mn0) time.