Minimum Cost Flows

Chapter
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

Span Tree Minimum Cost Reverse Edge Minimum Cost Flow Residual Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahuja, R.K., Magnanti, T.L., and Orlin, J.B. [1993]: Network Flows. Prentice-Hall, Englewood Cliffs 1993MATHGoogle Scholar
  2. Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., and Schrijver, A. [1998]: Combinatorial Optimization. Wiley, New York 1998, Chapter 4Google 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 5Google Scholar
  5. Jungnickel, D. [2007]: Graphs, Networks and Algorithms. Third Edition. Springer, Berlin 2007, Chapters 10 and 11Google Scholar
  6. Lawler, E.L. [1976]: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York 1976, Chapter 4Google Scholar
  7. Ruhe, G. [1991]: Algorithmic Aspects of Flows in Networks. Kluwer Academic Publishers, Dordrecht 1991CrossRefMATHGoogle Scholar
  8. Skutella, M. [2009]: An introduction to network flows over time. In: Research Trends in Combinatorial Optimization (W.J. Cook, L. Lovász, J. Vygen, eds.), Springer, Berlin 2009, pp. 451–482Google Scholar
  9. Arkin, E.M., and Silverberg, E.B. [1987]: Scheduling jobs with fixed start and end times. Discrete Applied Mathematics 18 (1987), 1–8CrossRefMATHMathSciNetGoogle Scholar
  10. Armstrong, R.D., and Jin, Z. [1997]: A new strongly polynomial dual network simplex algorithm. Mathematical Programming 78 (1997), 131–148MATHMathSciNetGoogle Scholar
  11. 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
  12. Cunningham, W.H. [1976]: A network simplex method. Mathematical Programming 11 (1976), 105–116CrossRefMATHMathSciNetGoogle Scholar
  13. Dantzig, G.B. [1951]: Application of the simplex method to a transportation problem. In: Activity Analysis and Production and Allocation (T.C. Koopmans, Ed.), Wiley, New York 1951, pp. 359–373Google Scholar
  14. Edmonds, J., and Karp, R.M. [1972]: Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM 19 (1972), 248–264CrossRefMATHGoogle Scholar
  15. Fleischer, L., and Skutella, M. [2007]: Quickest flows over time. SIAM Journal on Computing 36 (2007), 1600–1630CrossRefMATHMathSciNetGoogle Scholar
  16. Ford, L.R., and Fulkerson, D.R. [1958]: Constructing maximal dynamic flows from static flows. Operations Research 6 (1958), 419–433CrossRefMathSciNetGoogle Scholar
  17. Ford, L.R., and Fulkerson, D.R. [1962]: Flows in Networks. Princeton University Press, Princeton 1962MATHGoogle Scholar
  18. Gale, D. [1957]: A theorem on flows in networks. Pacific Journal of Mathematics 7 (1957), 1073–1082MATHMathSciNetGoogle Scholar
  19. Goldberg, A.V., and Tarjan, R.E. [1989]: Finding minimum-cost circulations by cancelling negative cycles. Journal of the ACM 36 (1989), 873–886CrossRefMATHMathSciNetGoogle Scholar
  20. Goldberg, A.V., and Tarjan, R.E. [1990]: Finding minimum-cost circulations by successive approximation. Mathematics of Operations Research 15 (1990), 430–466CrossRefMATHMathSciNetGoogle Scholar
  21. 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
  22. Hassin, R. [1983]: The minimum cost flow problem: a unifying approach to dual algorithms and a new tree-search algorithm. Mathematical Programming 25 (1983), 228–239CrossRefMATHMathSciNetGoogle Scholar
  23. Hitchcock, F.L. [1941]: The distribution of a product from several sources to numerous localities. Journal of Mathematical Physics 20 (1941), 224–230MathSciNetGoogle Scholar
  24. 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. 113–128Google Scholar
  25. Hoppe, B., and Tardos, É. [2000]: The quickest transshipment problem. Mathematics of Operations Research 25 (2000), 36–62CrossRefMATHMathSciNetGoogle Scholar
  26. Iri, M. [1960]: A new method for solving transportation-network problems. Journal of the Operations Research Society of Japan 3 (1960), 27–87Google Scholar
  27. Jewell, W.S. [1958]: Optimal flow through networks. Interim Technical Report 8, MIT 1958Google Scholar
  28. Karzanov, A.V., and McCormick, S.T. [1997]: Polynomial methods for separable convex optimization in unimodular linear spaces with applications. SIAM Journal on Computing 26 (1997), 1245–1275CrossRefMATHMathSciNetGoogle Scholar
  29. Klein, M. [1967]: A primal method for minimum cost flows, with applications to the assignment and transportation problems. Management Science 14 (1967), 205–220CrossRefMATHGoogle Scholar
  30. Klinz, B., and Woeginger, G.J. [2004]: Minimum cost dynamic flows: the series-parallel case. Networks 43 (2004), 153–162CrossRefMATHMathSciNetGoogle Scholar
  31. Orden, A. [1956]: The transshipment problem. Management Science 2 (1956), 276–285CrossRefMATHMathSciNetGoogle Scholar
  32. Ore, O. [1956]: Studies on directed graphs I. Annals of Mathematics 63 (1956), 383–406CrossRefMathSciNetGoogle Scholar
  33. Orlin, J.B. [1993]: A faster strongly polynomial minimum cost flow algorithm. Operations Research 41 (1993), 338–350CrossRefMATHMathSciNetGoogle Scholar
  34. Orlin, J.B. [1997]: A polynomial time primal network simplex algorithm for minimum cost flows. Mathematical Programming 78 (1997), 109–129MATHMathSciNetGoogle Scholar
  35. Orlin, J.B., Plotkin, S.A., and Tardos, É. [1993]: Polynomial dual network simplex algorithms. Mathematical Programming 60 (1993), 255–276CrossRefMATHMathSciNetGoogle Scholar
  36. 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
  37. 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. 473–483Google Scholar
  38. Schulz, A.S., and Weismantel, R. [2002]: The complexity of generic primal algorithms for solving general integer problems. Mathematics of Operations Research 27 (2002), 681–692CrossRefMATHMathSciNetGoogle Scholar
  39. Tardos, É. [1985]: A strongly polynomial minimum cost circulation algorithm. Combinatorica 5 (1985), 247–255CrossRefMATHMathSciNetGoogle Scholar
  40. Tolstoĭ, A.N. [1930]: Metody nakhozhdeniya naimen’shego summovogo kilometrazha pri planirovanii perevozok v prostanstve. In: Planirovanie Perevozok, Sbornik pervyĭ, Transpechat’ NKPS, Moskow 1930, pp. 23–55. (See A. Schrijver, On the history of the transportation and maximum flow problems, Mathematical Programming 91 (2002), 437–445)Google Scholar
  41. Tomizawa, N. [1971]: On some techniques useful for solution of transportation network problems. Networks 1 (1971), 173–194CrossRefMATHMathSciNetGoogle Scholar
  42. Vygen, J. [2002]: On dual minimum cost flow algorithms. Mathematical Methods of Operations Research 56 (2002), 101–126CrossRefMATHMathSciNetGoogle Scholar
  43. Wagner, H.M. [1959]: On a class of capacitated transportation problems. Management Science 5 (1959), 304–318CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany

Personalised recommendations