Maximium Priority Matchings

Let G = (V,E) be an undirected graph with n vertices and m edges, in which each vertex u is assigned an integer priority in [1, n], with 1 being the “highest” priority. Let M be a matching of G. We define the priority score of M to be an n-ary integer in which the i-th mostsignificant digit is the number of vertices with priority i that are incident to an edge in M . We describe a variation of the augmenting path method (Edmonds’ algorithm) that finds a matching with maximum priority score in O(mn) time.