The Dynamics of Rank-Maximal and Popular Matchings

Given a bipartite graph, where the two sets of vertices are applicants and posts and ranks on the edges represent preferences of applicants over posts, a rank-maximal matching is one in which the maximum number of applicants is matched to their rank one posts and subject to this condition, the maximum number of applicants is matched to their rank two posts, and so on. A rank-maximal matching can be computed in O(min(c √ n, n)m) time, where n denotes the number of applicants, m the number of edges and c the maximum rank of an edge in an optimal solution [9]. We study the dynamic version of the problem in which a new applicant or post may be added to the graph and we would like to maintain a rank-maximal matching. We show that after the arrival of one vertex, we are always able to update the existing rank-maximal matching in O(min(cn, n) + m) time; moreover, by the application of just one alternating path. The time bound can be considered optimal under the circumstances, as improving it would imply a better running time for the rank-maximal matching problem. Additionally, our solution has the property that enables to minimize the number of needed changes. As a by-product we also get an analogous O(m) result for the dynamic version of the (one-sided) popular matching problem.