Dynamic Rank-Maximal 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 \(G=(\mathcal {A}\cup \mathcal {P},E)\), 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.