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Polynomial-time primal simplex algorithms for the minimum cost network flow problem

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Abstract

We present two variants of the primal network simplex algorithm which solve the minimum cost network flow problem in at mostO(n 2 logn) pivots. Here we define the network simplex method as a method which proceeds from basis tree to adjacent basis tree regardless of the change in objective function value; i.e., the objective function is allowed to increase on some iterations. The first method is an extension of theminimum mean augmenting cycle-canceling method of Goldberg and Tarjan. The second method is a combination of a cost-scaling technique and a primal network simplex method for the maximum flow problem. We also show that the diameter of the primal network flow polytope is at mostn 2 m.

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References

  1. Ahuja, R. K., and J. B. Orlin. 1988. Improved Primal Simplex Algorithms for Shortest Path, Assignment and Minimum Cost Flow Problems. Working Paper No. 2090-882090-88, Sloan School of Management, M.I.T., Cambridge, MA.

    Google Scholar 

  2. Akgul, M. 1985. A Genuinely Polynomial Primal Simplex Algorithm for the Assignment Problem. Research Report, Department of Computer Science and Operations Research, North Carolina State University, Raleigh, NC.

    Google Scholar 

  3. Akgul, M. 1986. Shortest Paths and the Simplex Method. Technical Report, Department of Computer Sciences and Opterations Research Program, North Carolina State University, Raleigh, NC.

    Google Scholar 

  4. Ali, A., R. Helgason, J. Kennington, and H. Lall. 1978. Primal Network Simplex Codes: State of the Art Implementation Technology.Networks 8, 315–359.

    Article  MathSciNet  Google Scholar 

  5. Balinski, M. L. 1985. Signature Methods for the Assignment Problem.Oper. Res. 33, 527–536.

    MATH  MathSciNet  Google Scholar 

  6. Borgwardt, K. H. 1982. The Average Number of Pivot Steps Required by the Simplex Method is Polynomial.Z. Oper. Res. 26, 157–177.

    Article  MATH  MathSciNet  Google Scholar 

  7. Cunningham, W. H. 1979. Theoretical Properties of the Network Simplex Method.Math. Oper. Res. 4, 196–208.

    MATH  MathSciNet  Google Scholar 

  8. Dial, R., F. Glover, D. Karney, and D. Klingman. 1979. A Computational Analysis of Alternative Algorithms and Labeling Techniques for Finding Shortest Path Trees.Networks 9, 215–248.

    Article  MATH  MathSciNet  Google Scholar 

  9. Florian, M., S. Nguyen, and S. Pallottino. 1981. A Dual Simplex Algorithm for Finding All Shortest Paths.Networks 11, 367–378.

    Article  MATH  MathSciNet  Google Scholar 

  10. Goldberg, A. V., M. D. Grigoriadis, and R. E. Tarjan. 1991. Use of Dynamic Trees in a Network Simplex Algorithm for the Maximum Flow Problem.Programming 50, 277–290.

    Article  MATH  MathSciNet  Google Scholar 

  11. Goldberg, A. V., and R. E. Tarjan. 1989. Finding Minimum-Cost Circulations by Canceling Negative Cycles.J. Assoc. Comput. Mach. 36, 873–886.

    MATH  MathSciNet  Google Scholar 

  12. Goldberg, A. V., and R. E. Tarjan. 1990. Finding Minimum-Cost Circulations by Successive Approximations.Math. Oper. Res. 15, 430–466.

    MATH  MathSciNet  Google Scholar 

  13. Goldfarb, D. 1985. Efficient Dual Simplex Algorithms for the Assignment Problem.Math. Programming 33, 187–203.

    Article  MATH  MathSciNet  Google Scholar 

  14. Goldfarb, D., and J. Hao. 1990. A Primal Simplex Algorithm that Solves the Maximum Flow Problem in at mostnm Pivots andO(n 2 m) Time.Math. Programming 47, 353–365.

    Article  MATH  MathSciNet  Google Scholar 

  15. Goldfarb, D., J. Hao, and S. R. Kai. 1990. Efficient Shortest Path Simplex Algorithms.Oper. Res. 38, 624–628.

    MATH  MathSciNet  Google Scholar 

  16. Grigoriadis, M. D. 1986. An Efficient Implementation of the Network Simplex Method.Math. Programming Stud. 26, 83–111.

    MATH  MathSciNet  Google Scholar 

  17. Hung, M. S. 1983. A Polynomial Simplex Method for the Assignment Problem.Oper. Res. 31, 595–600.

    Article  MATH  MathSciNet  Google Scholar 

  18. Karp, R. M. 1978. A Characterization of the Minimum Cycle Mean in a Digraph,Discrete Math. 23, 309–311.

    MATH  MathSciNet  Google Scholar 

  19. Karp, R. M., and J. B. Orlin, 1981. Parametric Shortest Path Algorithms with an Application to Cyclic Staffing,Discrete Appl. Math. 3, 37–45.

    Article  MATH  MathSciNet  Google Scholar 

  20. Klein, M. 1967. A Primal Method for Minimal Cost Flow with Applications to the Assignment and Transportation Problems.Managment Sci. 14, 205–220.

    MATH  Google Scholar 

  21. Lawler, E. L. 1979. Shortest Path and Network Algorithms.Ann. Discrete Math. 4, 251–265.

    Article  MATH  MathSciNet  Google Scholar 

  22. Orlin, J. B. 1984. Genuinely Polynomial Simplex and Non-Simplex Algorithms for the Minimum Cost Flow Problem. Technical Report No. 1615-84, Sloan School of Management, M.I.T., Cambridge, MA.

    Google Scholar 

  23. Orlin, J. B. 1985. On the Simplex Algorithm for Networks and Generalized Networks.Math. Programming Stud. 25, 166–178.

    MathSciNet  Google Scholar 

  24. Rock, H. 1980. Scaling Techniques for Minimal Cost Network Flows. In: V. Page, ed.,Discrete Structures and Algorithms. Carl Hansen, Munich.

    Google Scholar 

  25. Roohy-Laleh, E. 1981. Improvements to the Theoretical Efficiency of the Simplex Method. Ph.D. Thesis, University of Carleton, Ottawa. Dissertation Abstracts International 13, 448B.

    Google Scholar 

  26. Smale, S. 1983. On the Average Number of Steps of the Simplex Method of Linear Programming.Math. Programming 27, 241–262.

    Article  MATH  MathSciNet  Google Scholar 

  27. Tardos, E. 1985. A Strongly Polynomial Minimum Cost Circulation Algorithm.Combinatorial 5, 247–255.

    Article  MATH  MathSciNet  Google Scholar 

  28. Tarjan, R. E. 1991. Efficiency of the Primal Network Simplex Algorithm for the Minimum-Cost Circulation Problem.Math. Oper. Res. 16, 272–291.

    MATH  MathSciNet  Google Scholar 

  29. Yemelichev V. A., M. M. Kovalev, and M. K. Kravtsov. 1984.Polytopes, Graphs and Optimisation (translated by G. H. Lawden), Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  30. Zadeh, N. 1979. Near Equivalence of Network Flow Algorithms. Technical Report No. 26, Dept. of Operations Research, Stanford University, CA.

    Google Scholar 

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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, 40 Sylvan Road, 02254, Waltham, MA, USA

    Jianxiu Hao

Authors
  1. Donald Goldfarb
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  2. Jianxiu Hao
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Additional information

Communicated by Nimrod Megiddo.

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

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Goldfarb, D., Hao, J. Polynomial-time primal simplex algorithms for the minimum cost network flow problem. Algorithmica 8, 145–160 (1992). https://doi.org/10.1007/BF01758840

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  • DOI: https://doi.org/10.1007/BF01758840

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