Testing popularity in linear time via maximum matching

Abstract

Popularity is an approach in mechanism design to find fair structures in a graph, based on the votes of the nodes. Popular matchings are the relaxation of stable matchings: given a graph $G=(V,E)$ with strict preferences on the neighbors of the nodes, a matching $M$ is popular if there is no other matching $M^{\prime }$ such that the number of nodes preferring $M^{\prime }$ is more than those preferring $M$. This paper considers the popularity testing problem, when the task is to decide whether a given matching is popular or not. Previous algorithms applied reductions to maximum weight matchings. We give a new algorithm for testing popularity by reducing the problem to maximum matching testing, thus attaining a linear running time $O\left(\right|E\left|\right)$.

Linear programming-based characterization of popularity is often applied for proving the popularity of a certain matching. As a consequence of our algorithm we derive a more structured dual witness than previous ones. Based on this result we give a combinatorial characterization of fractional popular matchings, which is a special class of popular matchings.