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The Ford–Fulkerson method or Ford–Fulkerson algorithm (FFA) is a greedy algorithm that computes the maximum flow in a flow network.It is sometimes called a "method" instead of an "algorithm" as the approach to finding augmenting paths in a residual graph is not fully specified [1] or it is specified in several implementations with different running times. [2]
Ford–Fulkerson algorithm: 1955 As long as there is an open path through the residual graph, send the minimum of the residual capacities on that path. Edmonds–Karp algorithm: 1970 A specialization of Ford–Fulkerson, finding augmenting paths with breadth-first search. Dinic's algorithm: 1970
Consider the flow f computed for G by Ford–Fulkerson algorithm. In the residual graph (G f ) obtained for G (after the final flow assignment by Ford–Fulkerson algorithm), define two subsets of vertices as follows: A: the set of vertices reachable from s in G f; A c: the set of remaining vertices i.e. V − A
This concept is used in Ford–Fulkerson algorithm which computes the maximum flow in a flow network. Note that there can be an unsaturated path (a path with available capacity) from u to v in the residual network, even though there is no such path from u to v in the original network.
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.
The Ford–Fulkerson algorithm, a greedy algorithm for maximum flow that is not in general strongly polynomial; The network simplex algorithm, a method based on linear programming but specialized for network flow [1]: 402–460 The out-of-kilter algorithm for minimum-cost flow [1]: 326–331
Minimum mean cycle canceling: a simple strongly polynomial algorithm. [5] Successive shortest path and capacity scaling: dual methods, which can be viewed as the generalization of the Ford–Fulkerson algorithm. [6] Cost scaling: a primal-dual approach, which can be viewed as the generalization of the push-relabel algorithm. [7]
The following is a longer example of mathematical-style pseudocode, for the Ford–Fulkerson algorithm: algorithm ford-fulkerson is input: Graph G with flow capacity c, source node s, sink node t output: Flow f such that f is maximal from s to t (Note that f (u,v) is the flow from node u to node v, and c (u,v) is the flow capacity from node u ...