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The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem. The maximum value of an s-t flow (i.e., flow from source s to sink t) is equal to the minimum capacity of an s-t cut (i.e., cut severing s from t) in the network, as stated in the max-flow min-cut theorem.
The flow network is at maximum flow if and only if it has a bottleneck with a value equal to zero. If any augmenting path exists, its bottleneck weight will be greater than 0. In other words, if there is a bottleneck value greater than 0, then there is an augmenting path from the source to the sink.
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]
In computer science and optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the total weight of the edges in a minimum cut, i.e., the smallest total weight of the edges which if removed would disconnect the source from the sink.
The max-flow min-cut theorem equates the value of a maximum flow to the value of a minimum cut, a partition of the vertices of the flow network that minimizes the total capacity of edges crossing from one side of the partition to the other. Approximate max-flow min-cut theorems provide an extension of this result to multi-commodity flow problems.
Dinic's algorithm or Dinitz's algorithm is a strongly polynomial algorithm for computing the maximum flow in a flow network, conceived in 1970 by Israeli (formerly Soviet) computer scientist Yefim Dinitz. [1]
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.
The concept of a preflow was originally designed by Alexander V. Karzanov and was published in 1974 in Soviet Mathematical Dokladi 15. This pre-flow algorithm also used a push operation; however, it used distances in the auxiliary network to determine where to push the flow instead of a labeling system.
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