Mathematical Programming

, Volume 5, Issue 1, pp 255–266 | Cite as

A bad network problem for the simplex method and other minimum cost flow algorithms

  • Norman Zadeh
Article

Abstract

For any integern, a modified transportation problem with 2n + 2 nodes is constructed which requires 2n + 2n−2−2 iterations using all but one of the most commonly used minimum cost flow algorithms.

As a result, the Edmonds—Karp Scaling Method [3] becomes the only known “good” (in the sense of Edmonds) algorithm for computing minimum cost flows.

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References

  1. [1]
    R.G. Busacker and P.J. Gowen, “A procedure for determining a family of minimal-cost network flow patterns”, Operations Research Office, The Johns Hopkins University, Baltimore, Md., No. ORO 15 (1961).Google Scholar
  2. [2]
    G.B. Dantzig,Linear programming and extensions (Princeton University Press, Princeton, N.J., 1962).Google Scholar
  3. [3]
    J. Edmonds and R.M. Karp, “Theoretical improvements in algorithmic efficiency for network flow problems”,Journal of the Association for Computing Machinery 19 (1972) 248–264.Google Scholar
  4. [4]
    L. Ford and D.R. Fulkerson, “A primal dual algorithm for the capacitated Hitchcock problem”,Naval Research Logistics Quarterly 4 (1957) 47–54.Google Scholar
  5. [5]
    L. Ford and D.R. Fulkerson,flows in networks (Princeton University Press, Princeton, N.J., 1962).Google Scholar
  6. [6]
    T.C. Hu,Integer programming and network flows (Addison-Wesley, Reading, Mass., 1969).Google Scholar
  7. [7]
    V.L. Klee and G.J. Minty, “How good is the simplex algorithm”, Boeing Math. Note No. 643 (February 1970).Google Scholar
  8. [8]
    M. Klein, “A primal method for minimal cost flows”,Management Science 14 (1967) 205–220.Google Scholar
  9. [9]
    N. Tomizawa, “On some techniques useful for solution of transportation network problems”,Networks 1 (1972) 173–194.Google Scholar
  10. [10]
    H.M. Wagner, “On a class of capacitated transportation problems”,Management Science 5 (1959) 304–318.Google Scholar
  11. [11]
    N. Zadeh, “Theoretical efficiency of the Edmonds—Karp algorithm for computing maximal flows”,Journal of the Association for Computing Machinery 19 (1972) 184–192.Google Scholar
  12. [12]
    N. Zadeh, “Theoretical efficiency and partial equivalence of minimum cost flow algorithms: A bad network problem for the simplex method”, Operations Research Center, University of California, Berkeley, Calif., No. ORC 72-7 (1972).Google Scholar
  13. [13]
    N. Zadeh, “More pathological examples for network flow problems”,Mathematical programming 5 (1973) 217–224.Google Scholar

Copyright information

© The Mathematical Programming Society 1973

Authors and Affiliations

  • Norman Zadeh
    • 1
  1. 1.IBM Thomas J. Watson Research CenterYorktown HeightsUSA

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