Skip to main content
SpringerLink
Log in

A primal simplex algorithm that solves the maximum flow problem in at mostnm pivots and O(n 2 m) time

  • Published:
Mathematical Programming Submit manuscript

Abstract

We propose a primal network simplex algorithm for solving the maximum flow problem which chooses as the arc to enter the basis one that isclosest to the source node from amongst all possible candidates. We prove that this algorithm requires at mostnm pivots to solve a problem withn nodes andm arcs, and give implementations of it which run in O(n 2 m) time. Our algorithm is, as far as we know, the first strongly polynomial primal simplex algorithm for solving the maximum flow problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.K. Ahuja and J.B. Orlin, “A fast and simple algorithm for the maximum flow problem,” Working Paper, Sloan School of Management, M.I.T. (Cambridge, MA, 1988).

    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 (1989), to appear.

  3. W.H. Cunningham, “A network simplex method,”Mathematical Programming 11 (1976) 105–116.

    Google Scholar 

  4. G.B. Dantzig, “Maximization of a linear function of variables subject to linear inequalities,” in: T.C. Koopmans, ed.,Activity Analysis of Production and Allocation (Wiley, New York, 1951) pp. 339–347.

    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 of Computing Machinery 19 (1972) 248–264.

    Google Scholar 

  7. L.R. Ford Jr. and D.R. Fulkerson, “A simple algorithm for finding maximal network flows and an application to the Hitchcock problem,”Canadian Journal of Mathematics 9 (1957) 210–218.

    Google Scholar 

  8. L.R. Ford Jr. and D.R. Fulkerson,Flows in Networks (Princeton University Press, Princeton, NJ, (1962).

    Google Scholar 

  9. A.V. Goldberg, “A new max-flow algorithm,” Technical report MIT/LCS/TM-291, Laboratory for Computer Science, M.I.T. (Cambridge, MA, 1985).

    Google Scholar 

  10. A.V. Goldberg, M.D. Grigoriadis and R.E. Tarjan, “Efficiency of the network simplex algorithm for the maximum flow problem,” Technical Report LCSR-TR-117, Laboratory for Computer Science Research, Rutgers University (New Brunswick, NJ, 1988).

    Google Scholar 

  11. A.V. Goldberg and R.E. Tarjan, “A new approach to the maximum flow problem,”18th Annual ACM Symposium on Theory of Computing (1986) 136–146.

  12. A.V. Goldberg and R.E. Tarjan, “Finding minimum-cost circulations by canceling negative cycles,” Technical Report MIT/LCS/TM-334, Laboratory for Computer Science, M.I.T. (Cambridge, MA, 1987).

    Google Scholar 

  13. D. Goldfarb, “Steepest-edge simplex algorithms for network flow problems,” Technical Report RC 6649, T.J. Watson Research Laboratory, IBM Corporation (Yorktown Heights, NY, 1977).

    Google Scholar 

  14. D. Goldfarb and M.D. Grigoriadis, “An efficient steepest-edge algorithm for the maximum flow problem,” presented atXth International Symposium on Mathematical Programming (Montreal, Que., 1979).

  15. D. Goldfarb and M.D. Grigoriadis, “A computational comparison of the Dinic and network simplex methods for maximum flow,” in: B. Simeone, P. Toth, G. Gallo, F. Maffioli and S. Pallottino, eds.,Fortran Codes for Network Optimization, Annals of Operations Research 13 (Baltzer, Basel, 1988).

    Google Scholar 

  16. D. Goldfarb and J.K. Reid, “A practicable steepest-edge simplex algorithm,”Mathematical Programming 12 (1977) 361–371.

    Google Scholar 

  17. A.V. Karzanov, “Determining the maximal flow in a network by the method of pre-flows,”Soviet Mathematics Doklady 15 (1974) 434–437.

    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. N. Zadeh, “A bad network problem for the simplex method and other minimum cost flow algorithms,”Mathematical Programming 5 (1973) 255–266.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial Engineering and Operations Research, Columbia University, 10027, New York, NY, USA

    Donald Goldfarb

  2. GTE Laboratories, 02254, Waltham, MA, USA

    Jianxiu Hao

Authors
  1. Donald Goldfarb
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianxiu Hao
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was supported in part by NSF Grants DMS 85-12277 and CDR 84-21402 and ONR Contract N00014-87-K0214.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goldfarb, D., Hao, J. A primal simplex algorithm that solves the maximum flow problem in at mostnm pivots and O(n 2 m) time. Mathematical Programming 47, 353–365 (1990). https://doi.org/10.1007/BF01580869

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01580869

Key words

Search

Navigation