Maximum Matchings and Popularity

TL;DR

This paper studies popular maximum matchings in bipartite graphs with preferences, providing a polynomial-time method to find minimum-cost popular max-matchings and showing NP-hardness for Pareto-optimal variants.

Contribution

It introduces a compact extended formulation for the popular max-matching polytope and demonstrates polynomial-time algorithms for min-cost popular max-matchings.

Findings

A compact extended formulation for the popular max-matching polytope.

Polynomial-time algorithm for min-cost popular max-matching.

NP-hardness of computing min-cost Pareto-optimal matchings.

Abstract

Let $G$ be a bipartite graph where every node has a strict ranking of its neighbors. For every node, its preferences over neighbors extend naturally to preferences over matchings. Matching $N$ is more popular than matching $M$ if the number of nodes that prefer $N$ to $M$ is more than the number that prefer $M$ to $N$. A maximum matching $M$ in $G$ is a "popular max-matching" if there is no maximum matching in $G$ that is more popular than $M$. Such matchings are relevant in applications where the set of admissible solutions is the set of maximum matchings and we wish to find a best maximum matching as per node preferences. It is known that a popular max-matching always exists in $G$. Here we show a compact extended formulation for the popular max-matching polytope. So when there are edge costs, a min-cost popular max-matching in $G$ can be computed in polynomial time. This is in contrast to the min-cost popular matching problem which is known to be NP-hard. We also consider Pareto-optimality, which is a relaxation of popularity, and show that computing a min-cost Pareto-optimal matching/max-matching is NP-hard.