Popular edges and dominant matchings by Ágnes CSEH and Telikepalli Kavitha was published in Mathematical programming.
Abstract
Given a bipartite graph G=(A∪B,E) with strict preference lists and given an edge e∗∈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(n3) algorithm to find a popular matching containing a given set of edges or report that none exists, where n=|A|+|B| .
