priority matching

{{short description|Graph matching with max number of high-priority vertices}}

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 {{math|1=G = (V, E)}}, and a partition of the vertex-set {{mvar|V}} into some {{mvar|k}} subsets, {{math|V{{sub|1}}, …, V{{sub|k}}}}, called priority classes. A priority matching is a matching that, among all possible matchings, saturates the largest number of vertices from {{math|V{{sub|1}}}}; subject to this, it saturates the largest number of vertices from {{math|V{{sub|2}}}}; subject to this, it saturates the largest number of vertices from {{math|V{{sub|3}}}}; and so on.

Priority matchings were introduced by Alvin Roth, Tayfun Sonmez and Utku Unver{{Cite journal|last1=Roth|first1=Alvin E.|last2=Sönmez|first2=Tayfun|last3=Utku Ünver|first3=M.|date=2005-12-01|title=Pairwise kidney exchange|journal=Journal of Economic Theory|language=en|volume=125|issue=2|pages=151–188|doi=10.1016/j.jet.2005.04.004|s2cid=583399|issn=0022-0531|url=http://papers.nber.org/papers/w10698.pdf }} 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{{Cite journal|last=Okumura|first=Yasunori|date=2014-11-01|title=Priority matchings revisited|journal=Games and Economic Behavior|language=en|volume=88|pages=242–249|doi=10.1016/j.geb.2014.10.007|issn=0899-8256}} 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 {{math|O({{abs|V}}{{abs|E}} + {{abs|V}}{{sup|2}} log {{abs|V}})}}.

Jonathan S. Turner{{cite arXiv|last=Turner|first=Jonathan|date=2015-12-28|title=Maximium [sic] Priority Matchings|class=cs.DS|eprint=1512.08555}} presented a variation of the augmenting path method (Edmonds' algorithm) that finds a priority matching in time {{math|O({{abs|V}}{{abs|E}})}}. Later, he found a faster algorithm for bipartite graphs: the algorithm runs in time{{cite arXiv|last=Turner|first=Jonathan|date=2015-12-31|title=Faster Maximium [sic] Priority Matchings in Bipartite Graphs|class=cs.DS|eprint=1512.09349}}

:$O(k\; |E|\; \backslash sqrt$

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