The following table gives a list of algorithms that can solve the maximum flow problem.

The following table lists algorithms for solving the maximum flow problem.

A new approach to the maximum flow problem.

For the example flow network in maximum flow problem we do the following:

Maximum flow problems can be solved efficiently with the Ford-Fulkerson algorithm.

Minimum cost maximum flow problem - First find the maximum flow amount .

By the max-flow min-cut theorem, one can solve the problem as a maximum flow problem.

The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem.

In this expanded network, the vertex capacity constraint is removed and therefore the problem can be treated as the original maximum flow problem.

Another application of widest paths arises in the Ford-Fulkerson algorithm for the maximum flow problem.

The maximum flow problem is to maximize , that is, to route as much flow as possible from to .

To find the maximum flow across , we can transform the problem into the maximum flow problem in the original sense by expanding .

The maximum flow problem was first formulated in 1954 by T. E. Harris as a simplified model of Soviet railway traffic flow.

This matrix arises as the coefficient matrix of the constraints in the linear programming formulation of the maximum flow problem on the following network:

In optimization theory, maximum flow problems involve finding a feasible flow through a single-source, single-sink flow network that is maximum.

An improved algorithm will solve the maximum flow problem for every pair (u,v) where u is arbitrarily fixed while v varies over all vertices.

If V is the number of vertices in the graph, this simple algorithm would perform iterations of the Maximum flow problem, which can be solved in time.

Such energy minimization problems can be reduced to instances of the maximum flow problem in a graph (and thus, by the max-flow min-cut theorem, define a minimal cut of the graph).

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.

Ford's paper with D. R. Fulkerson on the maximum flow problem and the Ford-Fulkerson algorithm for solving it, published as a technical report in 1954 and in a journal in 1956, established the max-flow min-cut theorem.

But, articles such as demonstrate how a PRAM-like abstraction can be supported by the explicit multi-threading (XMT) paradigm and articles such as demonstrate that a PRAM algorithm for the maximum flow problem can provide strong speedups relative to the fastest serial program for the same problem.