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 containsi¾ź$$e^*$$. 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 toi¾ź$$M'$$. 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 called 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 containsi¾ź$$e^*$$. This allows us to design a linear time algorithm for the popular edge problem. When preference lists are complete, we show an $$On^3$$ 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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