Popular Matchings in the Stable Marriage Problem

The input is a bipartite graph G = (A boolean OR B, E) where each vertex u is an element of A boolean OR B ranks its neighbors in a strict order of preference. 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*. We consider the problem of computing a maximum cardinality popular matching in G. It is known that popular matchings always exist in such an instance G, however the complexity of computing a maximum cardinality popular matching was not known so far. 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(mn(0)) algorithm for computing a maximum cardinality popular matching in G, where m = vertical bar E vertical bar and n(0) = min(vertical bar A vertical bar, vertical bar B).