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The '''Hopcroft–Karp algorithm''' finds maximum cardinality [[matching]]s in [[bipartite graph]]s in <math>O(\sqrt{V} E)</math> time, where ''V'' is the number of vertices and ''E'' is the number of edges of the graph.<ref>John E. Hopcroft, Richard M. Karp: ''An <math>n^{5/2}</math> Algorithm for Maximum Matchings in Bipartite Graphs.'' SIAM J. Comput. 2(4), 225-231 (1973)</ref> <ref name=clrs>{{Introduction to Algorithms|2|chapter=26.5: The relabel-to-front algorithm|pages=pp. 696–697}}</ref> In the worst case of dense graphs, i.e., when <math>E=O(V^2)</math>, the worst-case time estimate is <math>O(V^{5/2})</math>.
The algorithm is an adaptation of the [[Edmonds-Karp algorithm]] for maximum flow, since bipartite matching is equivalent to finding the maximum (integer) flow if the vertices in each partition are considered sources (respectively sinks) of capacity 1. The [[Max-flow min-cut theorem|minimal cut]] associated with the flow is equivalent to the [[König's theorem (graph theory)|minimal vertex cover]] associated with the matching.
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