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The maximum flow problem was first formulated in 1954 by T. E. Harris and F. S. Ross as a simplified model of Soviet railway traffic flow. [1] [2] [3]In 1955, Lester R. Ford, Jr. and Delbert R. Fulkerson created the first known algorithm, the Ford–Fulkerson algorithm.
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]
This means all v ∈ V \ {s, t} have no excess flow, and with no excess the preflow f obeys the flow conservation constraint and can be considered a normal flow. This flow is the maximum flow according to the max-flow min-cut theorem since there is no augmenting path from s to t. [8] Therefore, the algorithm will return the maximum flow upon ...
This caused a lack of any known polynomial-time algorithm to solve the max flow problem in generic cases. Dinitz's algorithm and the Edmonds–Karp algorithm (published in 1972) both independently showed that in the Ford–Fulkerson algorithm, if each augmenting path is the shortest one, then the length of the augmenting paths is non-decreasing ...
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 The push–relabel maximum flow algorithm, one of the ...
In computer science, the Edmonds–Karp algorithm is an implementation of the Ford–Fulkerson method for computing the maximum flow in a flow network in (| | | |) time. The algorithm was first published by Yefim Dinitz in 1970, [1] [2] and independently published by Jack Edmonds and Richard Karp in 1972. [3]
An integral maximum flow of minimum cost can be found in polynomial time; see network flow problem. Every integral maximum flow in this network corresponds to a matching in which at most c i tasks are assigned to each agent i and at most d j agents are assigned to each task j (in the balanced case, exactly c i tasks are assigned to i and ...
The minimum cost variant of the multi-commodity flow problem is a generalization of the minimum cost flow problem (in which there is merely one source and one sink ). Variants of the circulation problem are generalizations of all flow problems. That is, any flow problem can be viewed as a particular circulation problem.