# A Simple Algorithm for Finding Maximal Network Flows and an Application to the Hitchcock Problem

```@article{Ford1957ASA,
title={A Simple Algorithm for Finding Maximal Network Flows and an Application to the Hitchcock Problem},
author={Lester Randolph Ford and Delbert Ray Fulkerson},
year={1957},
volume={9},
pages={210 - 218}
}```
• Published 1957
• Mathematics
The network-flow problem, originally posed by T. Harris of the Rand Corporation, has been discussed from various viewpoints in (1; 2; 7; 16). The problem arises naturally in the study of transportation networks; it may be stated in the following way. One is given a network of directed arcs and nodes with two distinguished nodes, called source and sink, respectively. All other nodes are called intermediate. Each directed arc in the network has associated with it a nonnegative integer, its flow…
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## References

SHOWING 1-10 OF 13 REFERENCES
Computation of maximal flows in networks
• Computer Science
• 1955
A simple computational method, based on the simplex algorithm of linear programming, is proposed for the following problem: connecting two given points by way of a number of intermediate points, where each link of the network has a number assigned to it representing its capacity.
Notes on Linear Programming: Part 1. The Generalized Simplex Method for Minimizing a Linear Form under Linear Inequality Restraints
• Mathematics
• 1954
Abstract : The determination of optimum solutions to systems of linear inequalities has assumed increasing importance as a tool for mathematical analysis of certain problems in economics, logistics,
On Representatives of Subsets
Let a set S of mn things be divided into m classes of n things each in two distinct ways, (a) and (b); so that there are m (a)-classes and m (b)-classes. Then it is always possible to find a set R of
On the Max Flow Min Cut Theorem of Networks.
• Mathematics
• 1955
Abstract : It is shown that Menger's theorem and the Max Flow Min Cut Theorem on networks are applications of the duality theorem of linear inequality theory.
A DECOMPOSITION THEOREM FOR PARTIALLY ORDERED SETS
Otherwise a and b are non-comparable. A subset S of P is independent if every two distinct elements of S are non-comparable. S is dependent if it contains two distinct elements which are comparable.
A combinatorial algorithm for the assignment problem. Issue 11 of Logistics Papers
• A combinatorial algorithm for the assignment problem. Issue 11 of Logistics Papers
• 1954
A theorem onflows in networks
• A theorem onflows in networks