# Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems

@article{Edmonds1972TheoreticalII, title={Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems}, author={Jack Edmonds and Richard M. Karp}, journal={Journal of the ACM (JACM)}, year={1972}, volume={19}, pages={248 - 264} }

This paper presents new algorithms for the maximum flow problem, the Hitchcock transportation problem, and the general minimum-cost flow problem. Upper bounds on the numbers of steps in these algorithms are derived, and are shown to compale favorably with upper bounds on the numbers of steps required by earlier algorithms. First, the paper states the maximum flow problem, gives the Ford-Fulkerson labeling method for its solution, and points out that an improper choice of flow augmenting paths…

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## 2,410 Citations

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## References

SHOWING 1-10 OF 13 REFERENCES

On a Class of Capacitated Transportation Problems

- Mathematics
- 1959

Transportation models ordinary and transhipment having certain types of capacity constraints on the flows between origins and destinations are studied from the point of view of transforming them into…

Paths, Trees, and Flowers

- Mathematics
- 1965

A graph G for purposes here is a finite set of elements called vertices and a finite set of elements called edges such that each edge meets exactly two vertices, called the end-points of the edge. An…

Algorithm for solution of a problem of maximal flow in a network with power estimation

- Computer Science
- 1970

RECEIVED SEPTEMBER

- RECEIVED SEPTEMBER
- 1970

Bottleneck extrema

- 1968

On the equivalence of the capacity-constrained transshipment problem and the Hitchcock problems

- RAND Corp. Memorandum RM
- 1960

Apply Algorithm C repeatedly, starting with the pair (f, ~r), until a compatible pair (g, 8) is obtained. Set f = g and ~-= 0

By iteration of Algorithm C, the pair (2f

Note. References [1, 3, 6, 7] are not cited in the text

- Note. References [1, 3, 6, 7] are not cited in the text