# 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}, journal={Canadian Journal of Mathematics}, year={1957}, volume={9}, pages={210 - 218} }

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…

## 297 Citations

A COMPUTATIONAL METHOD FOR THE TRANSPORTATION PROBLEM ON A NETWORK

- Computer Science
- 2009

The author will give an extension of Ford-Fulkerson's primal-dual algorithm for the unc.apacitated or capacitated Hitchcock problem 4) to the transportation problem on a general network.

Flows, Paths, and Transportation 1. Flows

- Mathematics

In the 1950s, T.E. Harris at the RAND Corporation in Santa Monica (California) called attention for the following problem: Consider a rail network connecting two cities by way of a number of…

A Network-Flow Feasibility Theorem and Combinatorial Applications

- MathematicsCanadian Journal of Mathematics
- 1959

There are a number of interesting theorems, relative to capacitated networks, that give necessary and sufficient conditions for the existence of flows satisfying constraints of various kinds. Typical…

Network Flows

- Mathematics
- 2011

1 Defining Network Flow A flow network is a directed graph G = (V,E) in which each edge (u, v) ∈ E has non-negative capacity c(u, v) ≥ 0. We require that if (u, v) ∈ E, then (v, u) / ∈ E. That is, if…

Minimizing the maximum network flow: models and algorithms with resource synergy considerations

- Computer ScienceJ. Oper. Res. Soc.
- 2012

This paper proposes an inner-linearization procedure that significantly outperforms the competitive commercial solver SBB by improving the quality of solutions found by the latter by 6.2% (within a time limit of 1800 CPU s), while saving 84.5% of the required computational effort.

Ford-Fulkerson Max Flow Labeling Algorithm

- Computer Science
- 1998

The Ford-Fulkerson max ow labeling algorithm[3, 4] was introduced in the mid-1950's, and became the seminal work that is still applicable. The material presented in this note is taken from their…

Single-commodity robust network design problem: Complexity, instances and heuristic solutions

- Computer ScienceEur. J. Oper. Res.
- 2014

Multi-Commodity Network Flows

- Mathematics
- 1963

A network is a set of nodes Ni connected by arcs with nonnegative arc capacities bij which indicates the maximum amount of flow that can pass through the arc from Ni to Nj. Given all bij, there is a…

Matchings, Cuts, and Flows

- Computer Science
- 2014

This chapter describes a single algorithm with many variations and applications, that of finding maximum flow in a network, and the main solution strategy is the augmenting path method of Ford and Fulkerson.

## 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

- Mathematics
- 1935

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

- Mathematics
- 1950

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