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 {\em 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. 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 $\mathcal{O}(\min(c'n ,n^2) + m)$ 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. Additionally, we update the matching using a minimal number of changes (replacements). All cases of a deletion of a vertex/edge and an addition of an edge can be reduced to the problem of handling the addition of a vertex. As a by-product, we also get an analogous $\mathcal{O}(m)$ result for the dynamic version of the (one-sided) popular matching problem. Our results are based on the novel use of the properties of the Edmonds-Gallai decomposition. The presented ideas may find applications in other (dynamic) matching problems.