In graph theory, a priority matching (also called: maximum priority matching) is a matching that maximizes the number of high-priority vertices that participate in the matching. Formally, we are given a graph, and a partition of the vertex-set into some subsets,, called priority classes. A priority matching is a matching that, among all possible matchings, saturates the largest number of vertices from ; subject to this, it saturates the largest number of vertices from ; subject to this, it saturates the largest number of vertices from ; and so on.
Priority matchings were introduced by Alvin Roth, Tayfun Sonmez and Utku Unver[1] in the context of kidney exchange. In this problem, the vertices are patient-donor pairs, and each edge represents a mutual medical compatibility. For example, an edge between pair 1 and pair 2 indicates that donor 1 is compatible with patient 2 and donor 2 is compatible with patient 1. The priority classes correspond to medical priority among patients. For example, some patients are in a more severe condition so they must be matched first. Roth, Sonmez and Unver assumed that each priority-class contains a single vertex, i.e., the priority classes induce a total order among the pairs.
Later, Yasunori Okumura[2] extended the work to priority-classes that may contain any number of vertices. He also showed how to find a priority matching efficiently using an algorithm for maximum-cardinality matching, with a run-time complexity of .
Jonathan S. Turner[3] presented a variation of the augmenting path method (Edmonds' algorithm) that finds a priority matching in time . Later, he found a faster algorithm for bipartite graphs: the algorithm runs in time[4]
O(k|E|\sqrt{|V|})