Dynamic rank-maximal and popular matchings

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 boolean OR 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. (ACM Trans Algorithms 2(4): 602-610, 2006), which takes time O(min((r + n, r root n)m). Our algorithm can also be used for maintaining a popular matching in the one-sided preference model in O(m + n) time, whenever one exists