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In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. [1] In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a network flow problem.
An augmenting path in a matching problem is closely related to the augmenting paths arising in maximum flow problems, paths along which one may increase the amount of flow between the terminals of the flow. It is possible to transform the bipartite matching problem into a maximum flow instance, such that the alternating paths of the matching ...
The minimum weight bipartite matching problem or assignment problem is to find a perfect matching M ⊆ E whose total weight is minimized. The idea is to reduce this problem to a network flow problem. Let G′ = (V′ = A ∪ B, E′ = E). Assign the capacity of all the edges in E′ to 1.
A matching in G' induces a schedule for F and obviously maximum bipartite matching in this graph produces an airline schedule with minimum number of crews. [29] As it is mentioned in the Application part of this article, the maximum cardinality bipartite matching is an application of maximum flow problem.
A blocking flow is an ... In networks that arise from the bipartite matching problem, ... Dinitz's algorithm for finding a maximum flow in a network. Ben-Gurion ...
The simplest way to compute a maximum cardinality matching is to follow the Ford–Fulkerson algorithm. This algorithm solves the more general problem of computing the maximum flow. A bipartite graph (X + Y, E) can be converted to a flow network as follows. Add a source vertex s; add an edge from s to each vertex in X.
Examples include biological or social networks, which contain hundreds, thousands and even billions of nodes in some cases (e.g. Facebook or LinkedIn). 1-planarity [1] 3-dimensional matching [2] [3]: SP1 Bandwidth problem [3]: GT40 Bipartite dimension [3]: GT18 Capacitated minimum spanning tree [3]: ND5
A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it, unless it is a source, which has only outgoing flow, or sink, which has only incoming flow. A network can be used to model traffic in a computer network, circulation with demands, fluids in pipes, currents in an electrical circuit, or ...