Skip to main content
SpringerLink
Log in

Max Flow and Min Cut with bounded-length paths: complexity, algorithms, and approximation

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

We consider the “flow on paths” versions of Max Flow and Min Cut when we restrict to paths having at most B arcs, and for versions where we allow fractional solutions or require integral solutions. We show that the continuous versions are polynomial even if B is part of the input, but that the integral versions are polynomial only when B ≤ 3. However, when B ≤ 3 we show how to solve the problems using ordinary Max Flow/Min Cut. We also give tight bounds on the integrality gaps between the integral and continuous objective values for both problems, and between the continuous objective values for the bounded-length paths version and the version allowing all paths. We give a primal–dual approximation algorithm for both problems whose approximation ratio attains the integrality gap, thereby showing that it is the best possible primal–dual approximation algorithm.

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. Abel U., Carstens H.G., Deuber W., Prömel H.J.: Time Dependent and Hypergraphic Networks. Technical Report, Fakultät für Mathematik, Universität Bielefeld. Part of this report appeared as: “On Hypergraphic Networks” in Operations Research Verfahren 32(1979), 1–4 (1978)

    Google Scholar 

  2. Baier, G.: Flows with Path Restrictions. Ph.D. thesis, TU Berlin (2003)

  3. Baier, G., Erlebach, T., Hall, A., Kolman, P., Köhler, E., Pangrac, O., Schilling, H., Skutella, M.: Length-bounded cuts and flows. To appear in Transactions on Algorithms; an extended abstract appears in Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) Proceedings of the 33rd International Colloquium on Automata, Languages and Programming (ICALP’06), LNCS 4051, Springer, Berlin, pp. 679–690 (2006)

  4. Ben-Ameur W.: Constrained length connectivity and survivable networks. Networks 36, 17–33 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bley, A. (1997) Node-Disjoint Length-Restricted Paths. Master’s thesis, TU Berlin

  6. Bley A.: On the complexity of vertex-disjoint length-restricted path problems. Comput. Complex. 12, 131–149 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boyles S.M., Exoo G.: A counterexample to a conjecture on paths of bounded length. J. Graph Theory 6, 205–209 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Coullard C.R., Gamble A.B., Liu J.: The K-walk polyhedron. In: Du, D.-Z., Sun, J. (eds) Advances in Optimization and Approximation, pp. 9–29. Kluwer, Norwell (1994)

    Google Scholar 

  9. Dahl G., Gouveia L.: On the directed hop-constrained shortest path problem. Oper. Res. Lett. 32, 15–22 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dahl G., Foldnes N., Gouveia L.: A note on hop-constrained walk polytopes. Oper. Res. Lett. 32, 345–349 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Martin R.K., Rardin R.L., Campbell B.A.: Polyhedral characterization of discrete dynamic programming. Oper. Res. 38, 127–138 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Exoo G.: On line disjoint paths of bounded length. Discret. Math. 44, 317–318 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fleischer, L., Skutella, M.: Quickest flows over time. SIAM J. Comput. 36, 1600–1630; an extended abstract appeared in IPCO 2002, LNCS 2337, Springer, pp. 36–53 (2007)

  14. Ford L.R. Jr, Fulkerson D.R.: Maximal flow through a network. Can. J. Math. 8, 399–404 (1956)

    MATH  MathSciNet  Google Scholar 

  15. Fortz B., Mahjoub A.R., McCormick S.T., Pesneau P.: The 2-edge connected subgraph problem with bounded rings. Math. Prog. 105, 85–111 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Garey M.R., Johnson D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)

    MATH  Google Scholar 

  17. Gouveia L., Magnanti T.L.: Network flow models for designing diameter-constrained minimum-spanning and Steiner trees. Networks 41, 159–173 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gouveia L., Magnanti T.L., Requejo C.: A 2-path approach for odd-diameter-constrained minimum spanning and Steiner trees. Networks 44, 254–265 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Grötschel M., Lovász L., Schrijver A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)

    MATH  Google Scholar 

  20. Guruswami, V., Khanna, S., Rajaraman, R., Shepherd, B., Yannakakis, M.: (2003) Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems. J. Comput. Syst. Sci. 67, 473–496; a preliminary version appeared in Proceedings of 31st STOC, pp. 19–28 (1999)

    Google Scholar 

  21. Hoffman A.J.: A generalization of max flow- min cut. Math. Prog. 6, 352–359 (1974)

    Article  MATH  Google Scholar 

  22. Huygens D., Labbé M., Mahjoub A.R., Pesneau P.: The two-edge hop-constrained network design problem: valid inequalities and branch-and-cut. Networks 49, 116–133 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Huygens D., Mahjoub A.R.: Integer programming formulations for the two 4-hop-constrained paths problem. Networks 49, 135–144 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Huygens D., Mahjoub A.R., Pesneau P.: Two edge hop-constrained paths and polyhedra. SIAM J. Discret. Math. 18, 287–312 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Itai A., Perl Y., Shiloach Y.: The complexity of finding maximum disjoint paths with length constraints. Networks 12, 277–286 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kabadi, S.N., Kang, J., Chandrasekaran, R., Nair, K.P.K.: Hop Constrained Network Flows: Analysis and Synthesis. Technical report, University of New Brunswick (2004)

  27. Lovász L., Neumann-Lara V., Plummer M.: Mengerian theorems for paths of bounded length. Periodica Mathematica Hungarica 9, 269–276 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  28. Martens, M., McCormick, S.T.: A Polynomial Algorithm for Weighted Abstract Flow. Working paper, Sauder School of Business, UBC; an extended abstract appears in Proceedings of IPCO 2008, pp. 97–111 (2008)

  29. Maurras J-F., Vaxès Y.: Multicommodity netwok flow with jump constraints. Discret. Math. 165/166, 481–486 (1997)

    Article  Google Scholar 

  30. McCormick, S.T.: A Polynomial Algorithm for Abstract Maximum Flow. UBC Faculty of Commerce Working Paper 95-MSC-001. An extended abstract appears in Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 490–497 (1995)

  31. McCormick, S.T.: The Complexity of Max Flow and Min Cut with Bounded Length Paths. Talk at 6th International Workshop on Combinatorial Optimization, Aussois (2002)

  32. Pesneau, P.: Conception de Réseaux 2-Connexes avec Contraintes de Bornes. Ph.D. thesis, Université Blaise Pascal, Clermont-Ferrand (in French) (2003)

  33. Prömel H.J.: On Hypergraphic Networks. Universität Bielefeld, Diplomarbeit (1978)

    Google Scholar 

  34. Reiter, M.K., Stubblebine, S.G.: Path Independence for Authentication in Large-Scale Systems. AT&T Labs technical report (1996)

  35. Williamson, D.P.: Lecture Notes on Approximation Algorithms. IBM TJ Watson Research Center Technical Report RC 21409, Yorktown Heights (1998)

Download references

Author information

Authors and Affiliations

  1. Université Paris-Dauphine, LAMSADE, CNRS Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    A. Ridha Mahjoub

  2. Sauder School of Business, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    S. Thomas McCormick

Authors
  1. A. Ridha Mahjoub
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Thomas McCormick
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Ridha Mahjoub.

Additional information

A. Ridha Mahjoub: much of this work was done while at Laboratoire LIMOS, Université Blaise Pascal, Clermont-Ferrand.

S. Thomas McCormick: this research was partially supported by an NSERC Operating Grant, and a Visiting Professorship at Laboratoire LIMOS, Université Blaise Pascal, Clermont-Ferrand; a preliminary version of these results was [31].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mahjoub, A.R., McCormick, S.T. Max Flow and Min Cut with bounded-length paths: complexity, algorithms, and approximation. Math. Program. 124, 271–284 (2010). https://doi.org/10.1007/s10107-010-0366-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-010-0366-6

Mathematics Subject Classification (2000)

Search

Navigation