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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 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]
The maximum flow problem can be formulated as the maximization of the electrical current through a network composed of nonlinear resistive elements. [4] In this formulation, the limit of the current I in between the input terminals of the electrical network as the input voltage V in approaches ∞ {\displaystyle \infty } , is equal to the ...
Maximum flow problems can be solved in polynomial time with various algorithms (see table). The max-flow min-cut theorem states that finding a maximal network flow is equivalent to finding a cut of minimum capacity that separates the source and the sink, where a cut is the division of vertices such that the source is in one division and the ...
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 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.
The name "push–relabel" comes from the two basic operations used in the algorithm. Throughout its execution, the algorithm maintains a "preflow" and gradually converts it into a maximum flow by moving flow locally between neighboring nodes using push operations under the guidance of an admissible network maintained by relabel operations.
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 ...