# Finding minimum-cost flows by double scaling

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## Abstract

Several researchers have recently developed new techniques that give fast algorithms for the minimum-cost flow problem. In this paper we combine several of these techniques to yield an algorithm running in O(*nm*(log log*U*) log(*nC*)) time on networks with*n* vertices,*m* edges, maximum arc capacity*U*, and maximum arc cost magnitude*C.* The major techniques used are the capacity-scaling approach of Edmonds and Karp, the excess-scaling approach of Ahuja and Orlin, the cost-scaling approach of Goldberg and Tarjan, and the dynamic tree data structure of Sleator and Tarjan. For nonsparse graphs with large maximum arc capacity, we obtain a similar but slightly better bound. We also obtain a slightly better bound for the (noncapacitated) transportation problem. In addition, we discuss a capacity-bounding approach to the minimum-cost flow problem.

## Key words

Minimum-cost flows transportation problem scaling dynamic trees minimum-cost circulations## Preview

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## References

- [1]R.K. Ahuja and J.B. Orlin, “A fast and simple algorithm for the maximum flow problem,”
*Operations Research*37 (1989) 939–954.Google Scholar - [2]R.K. Ahuja, J.B. Orlin and R.E. Tarjan, “Improved time bounds for the maximum flow problem,”
*SIAM Journal on Computing*18 (1989) 939–954.Google Scholar - [3]D.P. Bertsekas, “A distributed algorithm for the assignment problem,” unpublished working paper, Laboratory for Information and Decision Science, M.I.T. (Cambridge, MA, 1979).Google Scholar
- [4]D.P. Bertsekas, “Distributed asynchronous relaxation methods for linear network flow problems,” Technical Report LIDS-P-1606, Laboratory for Information and Decision Sciences, M.I.T. (Cambridge, MA, 1986).Google Scholar
- [5]E.A. Dinic, “Algorithm for solution of a problem of maximum flow in networks with power estimation,”
*Soviet Mathematics Doklady*11 (1970) 1277–1280.Google Scholar - [6]J. Edmonds and R.M. Karp, “Theoretical improvements in algorithmic efficiency for network flow problems,”
*Journal of the Association on Computing Machinery*19 (1972) 248–264.Google Scholar - [7]H.N. Gabow, “Scaling algorithms for network problems,”
*Journal of Computer and System Sciences*31 (1985) 148–168.Google Scholar - [8]H.N. Gabow and R.E. Tarjan, “Faster scaling algorithms for networks problems,”
*SIAM Journal on Computing*18 (1989) 1013–1036.Google Scholar - [9]A.V. Goldberg, “Efficient graph algorithms for sequential and parallel computers,” Ph.D. Thesis, M.I.T. (Cambridge, MA, 1987).Google Scholar
- [10]A.V. Goldberg and R.E. Tarjan, “A new approach to the maximum flow problem,”
*Journal of the Association for Computing Machinery*35 (1988) 921–940.Google Scholar - [11]A.V. Goldberg and R.E. Tarjan, “Finding minimum-cost circulations by successive approximation,”
*Mathematics of Operations Research*15 (1990) 430–466.Google Scholar - [12]A.V. Goldberg and R.E. Tarjan, “Finding minimum-cost circulations by canceling negative cycles,”
*Journal of the Association for Computing Machinery*36 (1989) 873–886; also*Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing*(1988), 388–397.Google Scholar - [13]E.L. Lawler,
*Combinatorial Optimization: Networks and Matroids*(Holt, Reinhart and Winston, New York, 1976).Google Scholar - [14]R. Melhorn,
*Data Structures and Algorithms, Vol. 1: Sorting and Searching*(Springer, Berlin, 1984).Google Scholar - [15]J.B. Orlin, “On the simplex algorithm for networks and generalized networks,”
*Mathematical Programming*23 (1985) 166–178.Google Scholar - [16]J. Orlin, “A faster strongly polynomial minimum cost flow algorithm,”
*Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing*(1988) 377–387.Google Scholar - [17]C.H. Papadimitriou and K. Steiglitz,
*Combinatorial Optimization: Algorithms and Complexity*(Prentice-Hall, Englewood Cliffs, N.J, 1982).Google Scholar - [18]D.D. Sleator and R.E. Tarjan, “A data structure for dynamic trees,”
*Journal of Computer and System Sciences*26 (1983) 362–391.Google Scholar - [19]D.D. Sleator and R.E. Tarjan, “Self-adjusting binary search trees,”
*Journal of the Association for Computing Machinery*32 (1985) 652–686.Google Scholar - [20]E. Tardos, “A strongly polynomial minimum cost circulation algorithm,”
*Combinatorica*5 (1985) 247–255.Google Scholar - [21]R.E. Tarjan,
*Data Structures and Network Algorithms*(Society for Industrial and Applied Mathematics, Philadelphia, PA, 1983).Google Scholar - [22]R.E. Tarjan and C.J. van Wyk, “An O(
*n*log log*n*)-time algorithm for triangulating a simple polygon,”*SIAM Journal on Computing*17 (1988) 143–178.Google Scholar - [23]H.M. Wagner, “On a class of capacitated transportation problems,”
*Management Science*5 (1959) 304–318.Google Scholar - [24]N. Zadeh, “A bad network flow problem for the simplex method and other minimum cost flow algorithms,”
*Mathematical Programming*5 (1973) 255–266.Google Scholar