Popular edges and dominant matchings
Mathematical Programming(2017)
摘要
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(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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关键词
Popular matching,Matching under preferences,Dominant matching,05C70,68W40,05C85
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