Dynamic Rank Maximal Matchings

TL;DR

This paper introduces a dynamic algorithm for maintaining rank-maximal matchings in bipartite graphs with changing vertices and edges, significantly improving efficiency over static recomputation methods.

Contribution

It presents a simple, efficient O(r(m+n))-time algorithm for updating rank-maximal matchings dynamically, outperforming existing static algorithms when the maximum rank is small.

Findings

The algorithm efficiently updates matchings after vertex or edge modifications.

It outperforms static recomputation when maximum rank r is small relative to n.

The approach is applicable in real-time matching scenarios with dynamic data.

Abstract

We consider the problem of matching applicants to posts where applicants have preferences over posts. Thus the input to our problem is a bipartite graph G = (A U P,E), where A denotes a set of applicants, P is a set of posts, and there are ranks on edges which denote the preferences of applicants over posts. A matching M in G is called rank-maximal if it matches the maximum number of applicants to their rank 1 posts, subject to this the maximum number of applicants to their rank 2 posts, and so on. We consider this problem in a dynamic setting, where vertices and edges can be added and deleted at any point. Let n and m be the number of vertices and edges in an instance G, and r be the maximum rank used by any rank-maximal matching in G. We give a simple O(r(m+n))-time algorithm to update an existing rank-maximal matching under each of these changes. When r = o(n), this is faster than recomputing a rank-maximal matching completely using a known algorithm like that of Irving et al., which takes time O(min((r + n, r*sqrt(n))m).