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Given a bipartite graph, where the two sets of vertices are applicants andposts 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 ismatched to their rank one posts and subject to this condition, the maximumnumber of applicants is matched to their rank two posts, and so on. We studythe dynamic version of the problem in which a new applicant or post may beadded to the graph and we would like to maintain a rank-maximal matching. Weshow that after the arrival of one vertex, we are always able to update theexisting 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 $c'$ themaximum rank of an edge in an optimal solution. Additionally, we update thematching using a minimal number of changes (replacements). All cases of adeletion of a vertex/edge and an addition of an edge can be reduced to theproblem of handling the addition of a vertex. As a by-product, we also get ananalogous $\mathcal{O}(m)$ result for the dynamic version of the (one-sided)popular matching problem.  Our results are based on the novel use of the properties of theEdmonds-Gallai decomposition. The presented ideas may find applications inother (dynamic) matching problems.

The Dynamics of Rank-Maximal and Popular Matchings