# Simple Generalized Maximum Flow Algorithms

## Abstract

We introduce a *gain-scaling* technique for the generalized maximum flow problem. Using this technique, we present three simple and intuitive polynomial-time combinatorial algorithms for the problem. Truemper’s augmenting path algorithm is one of the simplest combi- natorial algorithms for the problem, but runs in exponential-time. Our first algorithm is a polynomial-time variant of Truemper’s algorithm. Our second algorithm is an adaption of Goldberg and Tarjan’s preflow- push algorithm. It is the first polynomial-time preflow-push algorithm in generalized networks. Our third algorithm is a variant of the Fat-Path capacity-scaling algorithm. It is much simpler than Radzik’s variant and matches the best known complexity for the problem. We discuss practical improvements in implementation.

## Keywords

Generalize Maximum Residual Network Canonical Label Lossy Network Negative Cost Cycle## Preview

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## References

- 1.R. K. Ahuja, T. L. Magnanti, and J. B. Orlin.
*Network Flows: Theory, Algorithms, and Applications*. Prentice Hall, Englewood Cliffs, New Jersey, 1993.Google Scholar - 2.Edith Cohen and Nimrod Megiddo. New algorithms for generalized network flows.
*Math Programming*, 64:325–336, 1994.CrossRefMathSciNetGoogle Scholar - 3.F. Glover, J. Hultz, D. Klingman, and J. Stutz. Generalized networks: A fundamental computer based planning tool.
*Management Science*, 24:1209–1220, 1978.Google Scholar - 4.F. Glover and D. Klingman. On the equivalence of some generalized network flow problems to pure network problems.
*Math Programming*, 4:269–278, 1973.zbMATHCrossRefMathSciNetGoogle Scholar - 5.F. Glover, D. Klingman, and N. Phillips. Netform modeling and applications.
*Interfaces*, 20:7–27, 1990.Google Scholar - 6.A. V. Goldberg, S. A. Plotkin, and É Tardos. Combinatorial algorithms for the generalized circulation problem. Technical Report STAN-CS-88-1209, Stanford University, 1988.Google Scholar
- 7.A. V. Goldberg, S. A. Plotkin, and É Tardos. Combinatorial algorithms for the generalized circulation problem.
*Mathematics of Operations Research*, 16:351–379, 1991.zbMATHMathSciNetGoogle Scholar - 8.A. V. Goldberg and R. E. Tarjan. A new approach to the maximum flow problem.
*Journal of the ACM*, 35:921–940, 1988.zbMATHCrossRefMathSciNetGoogle Scholar - 9.A. V. Goldberg and R. E. Tarjan. Finding minimum-cost circulations by canceling negative cycles.
*Journal of the ACM*, 36:388–397, 1989.CrossRefMathSciNetGoogle Scholar - 10.A. V. Goldberg and R. E. Tarjan. Solving minimum cost flow problems by successive approximation.
*Mathematics of Operations Research*, 15:430–466, 1990.zbMATHMathSciNetGoogle Scholar - 11.D. Goldfarb and Z. Jin. A polynomial dual simplex algorithm for the generalized circulation problem. Technical report, Department of Industrial Engineering and Operations Research, Columbia University, 1995.Google Scholar
- 12.D. Goldfarb and Z. Jin. A faster combinatorial algorithm for the generalized circulation problem.
*Mathematics of Operations Research*, 21:529–539, 1996.zbMATHMathSciNetCrossRefGoogle Scholar - 13.D. Goldfarb, Z. Jin, and J. B. Orlin. Polynomial-time highest gain augmenting path algorithms for the generalized circulation problem.
*Mathematics of Operations Research*. To appear.Google Scholar - 14.W. S. Jewell. Optimal flow through networks with gains.
*Operations Research*, 10:476–499, 1962.zbMATHMathSciNetGoogle Scholar - 15.Anil Kamath and Omri Palmon. Improved interior point algorithms for exact and approximate solution of multicommodity flow problems. In
*Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms*, pages 502–511, 1995.Google Scholar - 16.S. Kapoor and P. M. Vaidya. Speeding up Karmarkar’s algorithm for multicommodity flows.
*Math Programming*. To appear.Google Scholar - 17.K. Mehlhorn and S. Näher. A platform for combinatorial and geometric computing.
*CACM*, 38(1):96–102, 1995. http://ftp.mpi-sb.mpg.de/LEDA/leda.html Google Scholar - 18.S. M. Murray.
*An interior point approach to the generalized flow problem with costs and related problems*. PhD thesis, Stanford University, 1993.Google Scholar - 19.K. Onaga. Dynamic programming of optimum flows in lossy communication nets.
*IEEE Trans. Circuit Theory*, 13:308–327, 1966.Google Scholar - 20.T. Radzik. Faster algorithms for the generalized network flow problem.
*Mathematics of Operations Research*. To appear.Google Scholar - 21.T. Radzik. Approximate generalized circulation. Technical Report 93-2, Cornell Computational Optimization Project, Cornell University, 1993.Google Scholar
- 22.K. Truemper. On max flows with gains and pure min-cost flows.
*SIAM J. Appl. Math*, 32:450–456, 1977.zbMATHCrossRefMathSciNetGoogle Scholar - 23.P. Tseng and D. P. Bertsekas. An
*∈*-relaxation method for separable convex cost generalized network flow problems. In*5th International Integer Programming and Combinatorial Optimization Conference*, 1996.Google Scholar