Rank Maximal Matchings -- Structure and Algorithms
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.
Cite
@article{arxiv.1409.4977,
title = {Rank Maximal Matchings -- Structure and Algorithms},
author = {Pratik Ghoshal and Meghana Nasre and Prajakta Nimbhorkar},
journal= {arXiv preprint arXiv:1409.4977},
year = {2014}
}