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arXiv · 1508.00614

Popular Edges and Dominant Matchings

Agnes Cseh·Telikepalli Kavitha
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Abstract

Given a bipartite graph G = (A u B, E) with strict preference lists and and edge 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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Agnes Cseh, Telikepalli Kavitha. 2015-08-03. Popular Edges and Dominant Matchings. https://arxiv.org/abs/1508.00614

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