Rank Maximal Matchings -- Structure and Algorithms

TL;DR

This paper introduces a switching graph characterization for rank-maximal matchings in bipartite graphs, enabling efficient algorithms for computing, counting, and analyzing the popularity of such matchings, with complexity results and approximation methods.

Contribution

It develops a novel switching graph framework for rank-maximal matchings, leading to efficient algorithms and complexity insights for related problems.

Findings

Efficient algorithm for computing rank-maximal pairs.

Proved counting rank-maximal matchings is #P-Complete.

Provided an FPRAS for counting rank-maximal matchings.

Abstract

Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P denotes a set of posts and ranks on the edges denote preferences of the agents over posts. A matching M in G is rank-maximal if it matches the maximum number of applicants to their top-rank post, subject to this, the maximum number of applicants to their second rank post and so on. In this paper, we develop a switching graph characterization of rank-maximal matchings, which is a useful tool that encodes all rank-maximal matchings in an instance. The characterization leads to simple and efficient algorithms for several interesting problems. In particular, we give an efficient algorithm to compute the set of rank-maximal pairs in an instance. We show that the problem of counting the number of rank-maximal matchings is #P-Complete and also give an FPRAS for the problem. Finally, we consider the problem of deciding whether a rank-maximal matching is popular among all the rank-maximal matchings in a given instance, and give an efficient algorithm for the problem.