Skip to main content
SpringerLink
Log in

Algorithms and complexity analysis for some flow problems

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Several network-flow problems with additional constraints are considered. They are all special cases of the linear-programming problem and are shown to be ℘-complete. It is shown that the existence of a strongly polynomial-time algorithm for any of these problems implies the existence of such an algorithm for the general linear-programming problem. On the positive side, strongly polynomial algorithms for some parametric flow problems are given, when the number of parameters is fixed. These algorithms are applicable to constrained flow problems when the number of additional constraints is fixed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. M. Adel'son-Velskii, E. A. Dinic, and A. V. Karzanov.Flow Algorithms. Science, Moscow, 1975. In Russian.

    Google Scholar 

  2. C. Berge and A. Ghouila-Houri.Programming, Games and Transportation Networks. Wiley, New York, 1965.

    Google Scholar 

  3. P. J. Carstensen. The Complexity of Some Problems in Parametric, Linear, and Combinatorial Programming. Ph.D. thesis, Department of Mathematics, University of Michigan, Ann Arbor, MI, 1983.

    Google Scholar 

  4. E. Cohen. Combinatorial Algorithms for Optimization Problems. Ph.D. thesis, Department of Computer Science, Stanford University, Stanford, CA, 1991.

    Google Scholar 

  5. E. Cohen and N. Megiddo. Strongly polynomial and NC algorithms for detecting cycles in dynamic graphs.Proc. 21st Annual ACM Symposium on Theory of Computing, pp. 523–534. ACM, New York, 1989.

    Google Scholar 

  6. E. Cohen and N. Megiddo. Maximizing Concave Functions in Fixed Dimension. Technical Report RJ 7656 (71103), IBM Almaden Research Center, San Jose, CA 95120–6099, August 1990.

    Google Scholar 

  7. E. Cohen and N. Megiddo. Strongly polynomial time and NC algorithms for detecting cycles in periodic graphs.J. Assoc. Comput. Mach. To appear.

  8. E. A. Dinic. Algorithm for solution of a problem of maximum flow in networks with power estimation.Soviet Math. Dokl., 11:1277–1280, 1970.

    Google Scholar 

  9. D. P. Dobkin, R. J. Lipton, and S. P. Reiss. Linear programming is log-space hard for P.Inform. Process. Lett., 8(2):96–97, 1978.

    Google Scholar 

  10. D. P. Dobkin and S. P. Reiss. The complexity of linear programming.Theoret. Comput. Sci., 11:1–18, 1980.

    Google Scholar 

  11. J. Edmonds and R. M. Karp. Theoretical improvements in algorithmic efficiency for network flow problems.J. Assoc. Comput. Mach., 19:248–264, 1972.

    Google Scholar 

  12. L. R. Ford, Jr., and D. R. Fulkerson.Flows in Networks. Princeton University Press, Princeton, NJ, 1962.

    Google Scholar 

  13. G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications.SIAM J. Comput., 18:30–55, 1989.

    Google Scholar 

  14. A. Ghouila-Houri. Recherche du flot maximum dans certains réseaux lorsqu'on impose une condition de bouclage.Proc. 2nd International Conference on Operations Research, London, p. 156. American Mathematical Society, Providence, RI, 1960.

    Google Scholar 

  15. A. Ghouila-Houri. Une généralisation de l'algorithme de Ford-Fulkerson.C. R. Acad. Sci. Paris, 250:457, 1960.

    Google Scholar 

  16. A. V. Goldberg, É. Tardos, and R. E. Tarjan. Network Flow Algorithms. Technical Report STAN-CS-89-1252, Stanford University, 1989.

  17. P. Gordan. Über die auflösung linearer gleichungen mit reelen coefficienten.Math. Ann., 6:23–28, 1873.

    Google Scholar 

  18. D. Gusfield. Parametric combinatorial computing and a problem of program module distribution.J. Assoc. Comput. Mach., 30:551–563, 1983.

    Google Scholar 

  19. A. J. Hoffman. A generalization of max-flow min-cut.Math. Programming, 6:352–359, 1974.

    Google Scholar 

  20. A. Itai. Two-commodity flow.J. Assoc. Comput. Mach., 25(4):596–611, 1978.

    Google Scholar 

  21. E. L. Lawler.Combinatorial Optimization: Networks and Matroids. Holt, Reinhart, and Winston, New York, 1976.

    Google Scholar 

  22. N. Megiddo. A good algorithm for lexicographically optimal flows in multi-terminal networks.Bull. Amer. Math. Soc., 83:407–409, 1977.

    Google Scholar 

  23. N. Megiddo. Applying parallel computation algorithms in the design of serial algorithms.J. Assoc. Comput. Mach., 30:337–341, 1983.

    Google Scholar 

  24. N. Megiddo. Towards a genuinely polynomial algorithm for linear programming.SIAM J. Comput., 12:347–353, 1983.

    Google Scholar 

  25. N. Megiddo. Linear programming in linear time when the dimension is fixed.J. Assoc. Comput. Mach., 31:114–127, 1984.

    Google Scholar 

  26. C. H. Norton, S. A. Plotkin, and É. Tardos. Using separation algorithms in fixed dimension.Proc. 1st ACM-SIAM Symposium on Discrete Algorithms, pp. 377–387. ACM-SIAM, New York/Philadelphia, 1990.

    Google Scholar 

  27. É. Tardos. A strongly polynomial minimum cost circulation algorithm.Combinatorica, 5(3):247–255, 1985.

    Google Scholar 

  28. E. Tardos. A strongly polynomial algorithm to solve combinatorial linear programs.Oper. Res., 34:250–256, 1986.

    Google Scholar 

Download references

Author information

Author notes

  1. Edith Cohen

    Present address: Currently at AT&T Bell Laboratories, 600 Mountain Avenue, 07974, Murray Hill, NJ, USA

Authors and Affiliations

  1. IBM Almaden Research Center, 95120-6099, San Jose, CA, USA

    Nimrod Megiddo

  2. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Nimrod Megiddo

Authors
  1. Edith Cohen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nimrod Megiddo
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Harold N. Gabow.

Work on the paper was done while at Stanford University and IBM Almaden Research Center. This research was partially supported by NSF PYI Grant CCR-8858097.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cohen, E., Megiddo, N. Algorithms and complexity analysis for some flow problems. Algorithmica 11, 320–340 (1994). https://doi.org/10.1007/BF01240739

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01240739

Key words

Search

Navigation