Popular Edges and Dominant Matchings
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.
Cite
@article{arxiv.1508.00614,
title = {Popular Edges and Dominant Matchings},
author = {Agnes Cseh and Telikepalli Kavitha},
journal= {arXiv preprint arXiv:1508.00614},
year = {2015}
}