Dynamic Rank-Maximal Matchings

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 $G=(\mathcal {A}\cup \mathcal {P},E)$, where $\mathcal {A}$ denotes a set of applicants, $\mathcal {P}$ 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. [13], which takes time $O(\min ((r+n,r\sqrt{n})m)$.

R.V. Arvind—Part of the work was done when the author was a summer intern at Chennai Mathematical Institute.