Popular Edges and Dominant Matchings

TL;DR

This paper introduces dominant matchings as a new subclass of popular matchings in bipartite graphs with strict preferences, providing a linear time algorithm for the popular edge problem and methods to test the stability of all popular matchings.

Contribution

It identifies dominant matchings as a key subclass and develops a linear time algorithm for the popular edge problem, enhancing understanding of popular matchings.

Findings

Existence of a linear time algorithm for the popular edge problem.

Dominant matchings help determine if all popular matchings are stable.

Every popular matching containing a specific edge is either stable or dominant.

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.