Abstract
Given a bipartite graph $$G = (A \cup B,E)$$ with strict preference lists and given an edge $$e^* \in E$$ , we ask if there exists a popular matching in G that contains $$e^*$$ . We call this the popular edge problem. A matching M is popular if there is no matching $$M'$$ such that the vertices that prefer $$M'$$ to M outnumber those that prefer M to $$M'$$ . It is known that every stable matching is popular; however G may have no stable matching with the edge $$e^*$$ . In this paper we identify another natural subclass of popular matchings called “dominant matchings” and show that if there is a popular matching that contains the edge $$e^*$$ , then there is either a stable matching that contains $$e^*$$ or a dominant matching that contains $$e^*$$ . This allows us to design a linear time algorithm for identifying the set of popular edges. When preference lists are complete, we show an $$O(n^3)$$ algorithm to find a popular matching containing a given set of edges or report that none exists, where $$n = |A| + |B|$$ .
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