Popular Edges and Dominant Matchings
نویسندگان
چکیده
Given a bipartite graph G = (A∪B,E) with strict preference lists and e∗ ∈ E, we ask if there exists a popular matching in G that contains the edge 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 it. 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 the popular edge problem. We also use dominant matchings to efficiently test if every popular matching in G is stable or not.
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