Minimum Cost Flows

  • Bernhard Korte
  • Jens Vygen
Part of the Algorithms and Combinatorics book series (AC, volume 21)

Abstract

In this chapter we show how we can take edge costs into account. For example, in our application of the Maximum Flow Problem to the Job Assignment Problem mentioned in the introduction of Chapter 8 one could introduce edge costs to model that the employees have different salaries; our goal is to meet a deadline when all jobs must be finished at a minimum cost. Of course, there are many more applications.

Keywords

Transportation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

General Literature

  1. Ahuja, R.K., Magnanti, T.L., and Orlin, J.B. [1993]: Network Flows. Prentice-Hall, Engle-wood Cliffs 1993MATHGoogle Scholar
  2. Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., and Schrijver, A. [1998]: Combinatorial Optimization. Wiley, New York 1998, Chapter 4MATHGoogle Scholar
  3. Goldberg, A.V., Tardos, É., and Tarjan, R.E. [1990]: Network flow algorithms. In: Paths, Flows, and VLSI-Layout (B. Korte, L. Lovász, H.J. Prömel, A. Schrijver, eds.), Springer, Berlin 1990, pp. 101–164Google Scholar
  4. Gondran, M., and Minoux, M. [1984]: Graphs and Algorithms. Wiley, Chichester 1984, Chapter 5MATHGoogle Scholar
  5. Jungnickel, D. [1999]: Graphs, Networks and Algorithms. Springer, Berlin 1999, Chapter 9CrossRefGoogle Scholar
  6. Lawler, E.L. [1976]: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York 1976, Chapter 4MATHGoogle Scholar
  7. Ruhe, G. [1991]: Algorithmic Aspects of Flows in Networks. Kluwer Academic Publishers, Dordrecht 1991MATHCrossRefGoogle Scholar

Cited References

  1. Armstrong, R.D., and Jin, Z. [1997]: A new strongly polynomial dual network simplex algorithm. Mathematical Programming 78 (1997), 131–148MathSciNetMATHGoogle Scholar
  2. Busacker, R.G., and Gowen, P.J. [1961]: A procedure for determining a family of minimum-cost network flow patterns. ORO Technical Paper 15, Operational Research Office, Johns Hopkins University, Baltimore 1961Google Scholar
  3. Edmonds, J., and Karp, R.M. [1972]: Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM 19 (1972), 248–264MATHCrossRefGoogle Scholar
  4. Ford, L.R., and Fulkerson, D.R. [1962]: Flows in Networks. Princeton University Press, Princeton 1962MATHGoogle Scholar
  5. Gale, D. [1957]: A theorem on flows in networks. Pacific Journal of Mathematics 7 (1957), 1073–1082MathSciNetMATHCrossRefGoogle Scholar
  6. Goldberg, A.V., and Tarjan, R.E. [1989]: Finding minimum-cost circulations by cancelling negative cycles. Journal of the ACM 36 (1989), 873–886MathSciNetMATHCrossRefGoogle Scholar
  7. Grötschel, M., and Lovász, L. [1995]: Combinatorial optimization. In: Handbook of Combinatorics; Vol. 2 (R.L. Graham, M. Grötschel, L. Lovász, eds.), Elsevier, Amsterdam 1995Google Scholar
  8. Hassin, R. [1983]: The minimum cost flow problem: a unifying approach to dual algorithms and a new tree-search algorithm. Mathematical Programming 25 (1983), 228–239MathSciNetMATHCrossRefGoogle Scholar
  9. Hitchcock, F.L. [1941]: The distribution of a product from several sources to numerous localities. Journal of Mathematical Physics 20 (1941), 224–230MathSciNetGoogle Scholar
  10. Hoffman, A.J. [1960]: Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In: Combinatorial Analysis (R.E. Bellman, M. Hall, eds.), AMS, Providence 1960, pp. 113–128CrossRefGoogle Scholar
  11. Iri, M. [1960]: A new method for solving transportation-network problems. Journal of the Operations Research Society of Japan 3 (1960), 27–87Google Scholar
  12. Jewell, W.S. [1958]: Optimal flow through networks. Interim Technical Report 8, MIT 1958Google Scholar
  13. Klein, M. [1967]: A primal method for minimum cost flows, with applications to the assignment and transportation problems. Management Science 14 (1967), 205–220MATHCrossRefGoogle Scholar
  14. Orden, A. [1956]: The transshipment problem. Management Science 2 (1956), 276–285MathSciNetMATHCrossRefGoogle Scholar
  15. Ore, O. [1956]: Studies on directed graphs I. Annals of Mathematics 63 (1956), 383–406MathSciNetCrossRefGoogle Scholar
  16. Orlin, J.B. [1993]: A faster strongly polynomial minimum cost flow algorithm. Operations Research 41 (1993), 338–350MathSciNetMATHCrossRefGoogle Scholar
  17. Orlin, J.B. [1997]: A polynomial time primal network simplex algorithm for minimum cost flows. Mathematical Programming 78 (1997), 109–129MathSciNetMATHGoogle Scholar
  18. Orlin, J.B., Plotkin, S.A., and Tardos, É. [1993]: Polynomial dual network simplex algorithms. Mathematical Programming 60 (1993), 255–276MathSciNetMATHCrossRefGoogle Scholar
  19. Plotkin, S.A., and Tardos, É. [1990]: Improved dual network simplex. Proceedings of the 1st Annual ACM-SIAM Symposium on Discrete Algorithms (1990), 367-376Google Scholar
  20. Schulz, A.S., Weismantel, R., and Ziegler, G.M. [1995]: 0/1-Integer Programming: optimization and augmentation are equivalent. In: Algorithms — ESA’ 95; LNCS 979 (P. Spirakis, ed.), Springer, Berlin 1995, pp. 473–483CrossRefGoogle Scholar
  21. Vygen, J. [2000]: On dual minimum cost flow algorithms. Proceedings of the 32nd Annual ACM Symposium on the Theory of Computing (2000), 117-125; to appear in Mathematical Methods of Operations ResearchGoogle Scholar
  22. Wagner, H.M. [1959]: On a class of capacitated transportation problems. Management Science 5 (1959), 304–318MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Bernhard Korte
    • 1
  • Jens Vygen
    • 1
  1. 1.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany

Personalised recommendations