Faster Maximium Priority Matchings in Bipartite Graphs

A maximum priority matching is a matching in an undirected graph that maximizes a priority score defined with respect to given vertex priorities. An earlier paper showed how to find maximum priority matchings in unweighted graphs. This paper describes an algorithm for bipartite graphs that is faster when the number of distinct priority classes is limited. For graphs with k distinct priority classes it runs in O(kmn) time, where n is the number of vertices in the graph and m is the number of edges. The maximum priority matching problem was introduced in [9]. In this problem, each vertex has an integer-valued priority and the objective is to find a matching that maximizes a priority score defined with respect to these values. The priority score is defined as the n-ary number in which the i-th most-significant digit is the number of matched vertices with priority i. The earlier paper described an algorithm for finding maximum priority matchings in O(mn) time that is based on Edmonds’ algorithm for the maximum size matching problem [2, 3, 6]. In a private communication, Tarjan observed that the 2-priority case for bipartite graphs could be solved in O(mn1/2) time using a recent algorithm for weighted matchings in bipartite graphs [1, 4, 7]. One assigns a weight of 0, 1 or 2 to each edge based on the number of high priority vertices it is incident to. A maximum weight matching of this graph matches the largest possible