An approximate max-flow min-cut relation for undirected multicommodity flow, with applications


In this paper, we prove the first approximate max-flow min-cut theorem for undirected multicommodity flow. We show that for a feasible flow to exist in a multicommodity problem, it is sufficient that every cut's capacity exceeds its demand by a factor ofO(logClogD), whereC is the sum of all finite capacities andD is the sum of demands. Moreover, our theorem yields an algorithm for finding a cut that is approximately minimumrelative to the flow that must cross it. We use this result to obtain an approximation algorithm for T. C. Hu's generalization of the multiway-cut problem. This algorithm can in turn be applied to obtain approximation algorithms for minimum deletion of clauses of a 2-CNF≡ formula, via minimization, and other problems. We also generalize the theorem to hypergraph networks; using this generalization, we can handle CNF≡ clauses with an arbitrary number of literals per clause.

This is a preview of subscription content, access via your institution.


  1. [1]

    F. Barahona: On via minimization,IEEE Trans. Circuits Syst. 37, 410–416.

  2. [2]

    F. Barahona, M. Grötschel, M. Jünger, andG. Reinelt: An application of combinatorial optimization to statistical physics and circuit layout design,Operations Research 36 (1988), 493–513.

    Google Scholar 

  3. [3]

    R.-W. Chen, Y. Kajitani, andS.-P. Chan: A graph-theoretic via minimization algorithm for two-layer printed circuit boards,IEEE Trans. Circuits Systems, CAS-30 (1983), 284–299.

    Google Scholar 

  4. [4]

    H. Choi, K. Nakajima, andC.S. Rim: Graph bipartization and via minimization,SIAM J. of Discrete Math. 2 (1989), 38–47.

    Google Scholar 

  5. [5]

    V. Chvátal: Tough graphs and Hamiltonian circuits,Discrete Mathematics 5 (1973), 215–228.

    Google Scholar 

  6. [6]

    E. Dahlhous, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour, andM. Yannakakis: The complexity of multiway cuts,Proc. 24th ACM Symposium on Theory of Computing (1992), 241–251.

  7. [7]

    P. Elias, A. Feinstein, andC.E. Shannon: A note on the maximum flow through a network,IRS Trans. Information Theory IT 2 (1956), 117–119.

    Google Scholar 

  8. [8]

    L. R. Ford, Jr., andD. R. Fulkerson:Flows in Networks, Princeton University Press, Princeton, New Jersey (1962).

    Google Scholar 

  9. [9]

    A. Frank: Packing paths, circuits, and cuts—a survey, in:Paths, Flows, and VLSI Layout, ed. B. Korte, L. Lovász, H.J. Prömel, A. Schrijver, Springer-Verlag (1990), 47–100.

  10. [10]

    N. Garg, V. V. Vazirani, andM. Yannakakis: Approximate max-flow min-(multi)cut theorems and their applications.Proc. 25th ACM Symposium on the Theory of Computing (1993), 698–707.

  11. [11]

    M. R. Garey, andD. S. Johnson:Computers and Intractability: A guide to the theory of NP-completeness, W. H. Freeman, San Francisco (1979).

    Google Scholar 

  12. [12]

    T. C. Hu: Multicommodity network flows,Operations Research 11, (1963), 344–360.

    Google Scholar 

  13. [13]

    D. S. Johnson: Approximation algorithms for combinatorial problems,Journal of Computer and System Sciences 9 (1974), 256–278.

    Google Scholar 

  14. [14]

    N. D. Jones, Y. E. Lien, andW. T. Lasser: New problems complete for nondeterministic log space,Math. Systems Theory,10 (1976), 1–17.

    Google Scholar 

  15. [15]

    P. Klein, S. Plotkin, andS. Rao: Planar graphs, multicommodity flow, and network decomposition.Proc. 25th ACM Symposium on the Theory of Computing (1991), 682–690.

  16. [16]

    P. Klein, S. Plotkin, S. Rao, andÉ. Tardos: Bounds on the max-flow min-cut ratio for directed multicommodity flows, Brown University Technical Report CS-93-30 (1993).

  17. [17]

    P. Klein, S. Plotkin, C. Stein, andÉ. Tardos: Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts,SIAM Journal on Computing, to appear.

  18. [18]

    F. T. Leighton, F. Makedon, S. Plotkin, C. Stein, É. Tardos, S. Tragoudas: Fast approximation algorithms for multicommodity flow problems,Proceedings of the 23rd Annual ACM Symposium on Theory of Computing (1991), 101–111.

  19. [19]

    F. T. Leighton, andS. Rao: An approximate max-flow min-cut theorem for uniform multicommodity flow problems with application to approximation algorithms,Proceedings, 29th Symposium on Foundations of Computer Science (1988), 422–431.

  20. [20]

    F. T. Leighton, F. Makedon, andS. Tragoudas: personal communication, 1990

  21. [21]

    P. Molitor: On the contact-minimization problem,Proc. 4th Annual Symposium on Theoretical Aspects of Computer Science (1987), published asLecture Notes in Computer Science 247, Springer-Verlag, New York, Berlin (1987), 420–431.

    Google Scholar 

  22. [22]

    N. Naclerio, S. Masuda, andK. Nakajima: Via minimization for gridless layouts,Proc. 24th ACM/IEEE Design Automation Conference (1987), 159–165.

  23. [23]

    N. Naclerio, S. Masuda, andK. Nakajima: The via minimization problem is NP-complete,IEEE Trans. Comput. 38, 1604–1608.

  24. [24]

    C. H. Papadimitriou, andM. Yannakakis: Optimization, approximation, and complexity classes,Proceedings, 20th ACM Symposium on Theory of Computing (1988), 229–234.

  25. [25]

    R. Pinter: Optimal layer assignment for interconnect,Journal of VLSI and Computer Systems 1 (1984), 123–137.

    Google Scholar 

  26. [26]

    D. Plaisted: A heuristic algorithm for small separators in arbitrary graphs,SIAM J. Comput. 19 (1990), 267–280.

    Google Scholar 

  27. [27]

    S. Plotkin, andÉ. Tardos: Improved bounds on the max-flow min-cut ratio for multicommodity flows.Proc. 25th ACM Symposium on the Theory of Computing (1993), 691–697.

  28. [28]

    S. Rao: Finding near optimal separators in planar graphs,Proceedings, 28th Annual Symposium on Foundations of Computer Science (1987), 225–237.

  29. [29]

    A. Schrijver: Min-max results in combinatorial optimization,Mathematical Programming, the state of the art. Springer-Verlag, Bonn (1983), 439–500.

    Google Scholar 

  30. [30]

    F. Shahrokhi, andD. Matula: The maximum concurrent flow problem,Journal of the ACM 37:2 (1990), 318–334

    Google Scholar 

  31. [31]

    S. Tragoudas: VLSI partitioning approximation algorithms based on multicommodity flow and other techniques, PhD thesis, University of Texas at Dallas (1991).

  32. [32]

    M. Yannakakis: Edge-Deletion problems,SIAM J. Computing 10, (1981), 297–309.

    Google Scholar 

Download references

Author information



Additional information

Most of the results in this paper were presented in preliminary form in “Approximation through multicommodity flow”,Proceedings, 31th Annual Symposium on Foundations of Computer Science (1990), pp. 726–737.

Research supported by the National Science Foundation under NSF grant CDA 8722809, by the Office of Naval and the Defense Advanced Research Projects Agency under contract N00014-83-K-0146, and ARPA Order No. 6320, Amendament 1.

Research supported by NSF grant CCR-9012357 and by an NSF Presidential Young Investigator Award.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Klein, P., Rao, S., Agrawal, A. et al. An approximate max-flow min-cut relation for undirected multicommodity flow, with applications. Combinatorica 15, 187–202 (1995).

Download citation

Mathematics Subject Classification (1991)

  • 68 Q 25
  • 68 R 10
  • 90 C 08
  • 90 C 27