Mathematical Programming

, Volume 53, Issue 1–3, pp 243–266 | Cite as

Finding minimum-cost flows by double scaling

  • Ravindra K. Ahuja
  • Andrew V. Goldberg
  • James B. Orlin
  • Robert E. Tarjan


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 logU) log(nC)) time on networks withn vertices,m edges, maximum arc capacityU, and maximum arc cost magnitudeC. 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 


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  1. [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. [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. [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    D.P. Bertsekas, “A distributed algorithm for the assignment problem,” unpublished working paper, Laboratory for Information and Decision Science, M.I.T. (Cambridge, MA, 1979).Google Scholar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. [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. [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. [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. [7]
    H.N. Gabow, “Scaling algorithms for network problems,”Journal of Computer and System Sciences 31 (1985) 148–168.Google Scholar
  8. [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. [9]
    A.V. Goldberg, “Efficient graph algorithms for sequential and parallel computers,” Ph.D. Thesis, M.I.T. (Cambridge, MA, 1987).Google Scholar
  10. [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. [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. [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; alsoProceedings of the Twentieth Annual ACM Symposium on Theory of Computing (1988), 388–397.Google Scholar
  13. [13]
    E.L. Lawler,Combinatorial Optimization: Networks and Matroids (Holt, Reinhart and Winston, New York, 1976).Google Scholar
  14. [14]
    R. Melhorn,Data Structures and Algorithms, Vol. 1: Sorting and Searching (Springer, Berlin, 1984).Google Scholar
  15. [15]
    J.B. Orlin, “On the simplex algorithm for networks and generalized networks,”Mathematical Programming 23 (1985) 166–178.Google Scholar
  16. [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. [17]
    C.H. Papadimitriou and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity (Prentice-Hall, Englewood Cliffs, N.J, 1982).Google Scholar
  18. [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. [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. [20]
    E. Tardos, “A strongly polynomial minimum cost circulation algorithm,”Combinatorica 5 (1985) 247–255.Google Scholar
  21. [21]
    R.E. Tarjan,Data Structures and Network Algorithms (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1983).Google Scholar
  22. [22]
    R.E. Tarjan and C.J. van Wyk, “An O(n log logn)-time algorithm for triangulating a simple polygon,”SIAM Journal on Computing 17 (1988) 143–178.Google Scholar
  23. [23]
    H.M. Wagner, “On a class of capacitated transportation problems,”Management Science 5 (1959) 304–318.Google Scholar
  24. [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

Copyright information

© The Mathematical Programming Society, Inc. 1992

Authors and Affiliations

  • Ravindra K. Ahuja
    • 1
  • Andrew V. Goldberg
    • 2
  • James B. Orlin
    • 1
  • Robert E. Tarjan
    • 3
    • 4
  1. 1.Sloan School of ManagementM.I.T.CambridgeUSA
  2. 2.Department of Computer ScienceStanford UniversityStanfordUSA
  3. 3.Department of Computer SciencePrinceton UniversityPrincetonUSA
  4. 4.NEC Research InstitutePrincetonUSA

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