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===Maximum Matching===

===Maximum Matching===

 

In graph theory, a [[matching (graph theory)|matching]] is a set of edges without common vertices. Liu et al.<ref name="Liu-Nature-11"/> extended this definition to directed graph, where a matching is a set of directed edges that do not share start or end vertices. It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical [[Hopcroft–Karp algorithm]], which runs in O(''E''{{radic|''N''}}) time in the worst case. For undirected graph, analytical solutions of the size and number of maximum matchings have been studied using the [[cavity method]] developed in statistical physics.<ref name="Zdeborova-06">[[Lenka Zdeborová|L. Zdeborová]] and M. Mezard, ''J. Stat. Mech.'' '''05''' (2006).</ref> Liu et al.<ref name="Liu-Nature-11"/> extended the calculations for directed graph.

In graph theory, a [[matching (graph theory)|matching]] is a set of edges without common vertices. Liu et al.<ref name="Liu-Nature-11"/> extended this definition to directed graph, where a matching is a set of directed edges that do not share start or end vertices. It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical [[Hopcroft–Karp algorithm]], which runs in O(''E''{{radic|''N''}}) time in the worst case. For undirected graph, analytical solutions of the size and number of maximum matchings have been studied using the [[cavity method]] developed in statistical physics.<ref name="Zdeborova-06">[[Lenka Zdeborová|L. Zdeborová]] and M. Mezard, ''J. Stat. Mech.'' '''05''' (2006).</ref> Liu et al.<ref name="Liu-Nature-11"/> extended the calculations for directed graph.