Advertisement

Minimum Cost Network Flow Algorithms

  • Jeffery L. Kennington
  • Richard V. Helgason

Abstract

The minimal cost network flow model is defined along with optimality criteria and three efficient procedures for obtaining an optimal solution. Primal and dual network simplex methods are specializations of well-known algorithms for linear programs. The primal procedure maintains primal feasibility at each iteration and seeks to simultaneously achieve dual feasibility, The dual procedure maintains dual feasibility and moves toward primal feasibility. All operations for both algorithms can be performed on a graphical structure called a tree. The scaling push-relabel method is designed exclusively for optimization problems on a network. Neither primal nor dual feasibility is achieved until the final iteration.

Keywords

Networks-graphs flow algorithms integer programming algorithms linear programming algorithms linear programming simplex algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. R. Ahuja, T. Magnanti, and J. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs, NJ 07632, 1993.Google Scholar
  2. R. Barr, F. Glover, and D. Klingman. Enhancement of spanning tree labeling procedures for network optimization. INFOR, 17:16–34, 1979.zbMATHGoogle Scholar
  3. D. Bertsekas. Linear Network Optimization: Algorithms and Codes. The MIT Press, Cambridge, MA, 1991.zbMATHGoogle Scholar
  4. D. Bertsekas and P. Tseng. RELAXT-III: A new and improved version of the RELAX code, lab. for information and decision systems report p-1990. Technical report, MIT, Cambridge, MA, 1990.Google Scholar
  5. D. Bertsekas and P. Tseng, RELAX-IV: A faster version of the RELAX code for solving minimum cost flow problems. Technical report, Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, 1994.Google Scholar
  6. R. Bland and D. Jensen. On the computational behavior of a polynomial-time network flow algorithm. Mathematical Programming, 54:1–43, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  7. G. Bradley, G. Brown, and G. Graves. Design and implementation of large-scale primal transshipment algorithms. Management Science, 21:1–38, 1977.CrossRefGoogle Scholar
  8. A. Charnes and W. Cooper. Management Models and Industrial Applications of Linear Programming: Volume I. John Wiley and Sons, Inc., New York, NY, 1967.Google Scholar
  9. G. Dantzig. Application of the simplex method to a transportation problem. In T. Koopmans, editor, Activity Analysis of Production and Allocation, pages 359–373. John Wiley and Sons, Inc., New York, NY, 1951.Google Scholar
  10. G. Dantzig. Linear Programming and Extensions. Princeton University Press, Princeton, NJ, 1963.zbMATHGoogle Scholar
  11. L. Ford and D. Fulkerson. Flows in Networks. Princeton University Press, Princeton, NJ, 1962.zbMATHGoogle Scholar
  12. F. Glover, D. Karney, and D. Klingman. Implementation and computational comparisons of primal, dual, and primal-dual computer codes for minimum cost network flow problems. Networks, 4:191–212, 1974a.zbMATHCrossRefGoogle Scholar
  13. F. Glover, D. Karney, D. Klingman, and A. Napier. A computational study on start procedures, basis change criteria, and solution algorithms for transportation problems. Management Science, 20:793–813, 1974b.zbMATHCrossRefMathSciNetGoogle Scholar
  14. F. Glover, D. Klingman, and J. Stutz. Augmented threaded index method for network optimization. INFOR, 12:293–298, 1974c.zbMATHGoogle Scholar
  15. A. Goldberg. An efficient implementation of a scaling minimum cost flow algorithm,. Technical Report STAT-CS-92-1439, Computer Science Department, Stanford University, Stanford, CA, 1992.Google Scholar
  16. A. Goldberg. An efficient implementation of a scaling minimum-cost flow algorithm. Journal of Algorithms, 22:1–29, 1997.CrossRefMathSciNetGoogle Scholar
  17. A. Goldberg and M. Kharitonov. On implementing scaling push-relabel algorithms for the minimum-cost flow problem. In D. Johnson and C. McGeoch, editors, Network Flows and Matching: First DIMACS Implementation Challenge, pages 157–198. AMS, Providence, RI, 1993.Google Scholar
  18. M. Grigoriadis. An efficient implementation of the network simplex method. Mathematical Programming Study, 26:83–111, 1986.zbMATHMathSciNetGoogle Scholar
  19. J. Kennington and R. Helgason. Algorithms for Network Programming. John Wiley and Sons, Inc., New York, NY, 1980.zbMATHGoogle Scholar
  20. J. Kennington and R. Mohamed. An efficient dual simplex optimizer for generalized networks. In R. Barr, R. Helgason, and J. Kennington, editors, Interfaces in Computer Science and Operations Research, pages 153–182. Kluwer Academic Publishers, Norwell, MA 02061, 1997.Google Scholar
  21. J. Kennington and J. Whitler, Simplex versus cost scaling algorithms for pure networks: An empirical analysis. Technical Report 96-CSE-8, Department of Computer Science and Engineering, Southern Methodist University, Dallas, TX, 1998.Google Scholar
  22. H. Röck. Scaling techniques for minimal cost network flows. In U. Pape, editor, Discrete Structures and Algorithms, pages 181–191. Carl Hanser, Munich, 1980.Google Scholar
  23. V. Srinivasan and G. Thompson. Benefit-cost analysis of coding techniques for the primal transportation algorithm. Journal of the Association for Computing Machinery, 20:194–213, 1973.zbMATHMathSciNetGoogle Scholar
  24. É. Tardos. A strongly polynomial minimum cost circulation algorithm. Combinatorica, 5:247–255, 1985.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Jeffery L. Kennington
    • 1
  • Richard V. Helgason
    • 1
  1. 1.Southern Methodist UniversityDallasUSA

Personalised recommendations