Rank-Maximal Matchings – Structure and Algorithms

Abstract

Let $G = (\mathcal {A}\cup \mathcal {P}, E)$ be a bipartite graph where $\mathcal {A}$ denotes a set of agents, $\mathcal {P}$ 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.

Meghana Nasre: Part of the work done by the author was supported by IIT-M initiation grant CSE/14-15/824/NFIG/MEGA.

Prajakta Nimbhorkar: Part of the work has been done while the author was on a sabbatical to the Institute of Mathematics of the Czech Academy of Sciences, Prague.