Priority matching

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 G = (V, E), and a partition of the vertex-set V into some k subsets, V1, ..., Vk, called priority classes. A priority matching is a matching that, among all possible matchings, saturates the largest number of vertices from V1; subject to this, it saturates the largest number of vertices from V2; subject to this, it saturates the largest number of vertices from V3; 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 O (|V| |E| + |V|2 log |V|).

Jonathan S. Turner[3] presented a variation of the augmenting path method (Edmonds' algorithm) that finds a priority matching in time O (|V| |E|). Later, he found a faster algorithm for bipartite graphs: the algorithm runs in time O(k |E| |V|1/2).[4]

See also

References
  1. Roth, Alvin E.; Sönmez, Tayfun; Utku Ünver, M. (2005-12-01). "Pairwise kidney exchange". Journal of Economic Theory. 125 (2): 151–188. doi:10.1016/j.jet.2005.04.004. ISSN 0022-0531. S2CID 583399.
  2. Okumura, Yasunori (2014-11-01). "Priority matchings revisited". Games and Economic Behavior. 88: 242–249. doi:10.1016/j.geb.2014.10.007. ISSN 0899-8256.
  3. Turner, Jonathan (2015-12-28). "Maximium Priority Matchings". arXiv:1512.08555 [cs.DS].
  4. Turner, Jonathan (2015-12-31). "Faster Maximium Priority Matchings in Bipartite Graphs". arXiv:1512.09349 [cs.DS].
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