Skip to main content
SpringerLink
Log in

The diameters of network-flow polytopes satisfy the Hirsch conjecture

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

Abstract

We solve a problem in the combinatorics of polyhedra motivated by the network simplex method. We show that the Hirsch conjecture holds for the diameter of the graphs of all network-flow polytopes, in particular the diameter of a network-flow polytope for a network with n nodes and m arcs is never more than \(m+n-1\). A key step to prove this is to show the same result for classical transportation polytopes.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Balinski, M.L.: The Hirsch conjecture for dual transportation polyhedra. Math. Oper. Res. 9(4), 629–633 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balinski, M.L., Russakoff, A.: On the assignment polytope. SIAM Rev. 16(4), 516–525 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonifas, N., Di Summa, M., Eisenbrand, F., Hähnle, N., Niemeier, M.: On sub-determinants and the diameter of polyhedra. Discrete Comput. Geom. 52, 102–115 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borgwardt, S.: On the diameter of partition polytopes and vertex-disjoint cycle cover. Math. Program. Ser. A 141, 1–20 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borgwardt, S. De Loera, J.A., Finhold E., Miller, J.: The hierarchy of circuit diameters and transportation polytopes. Discrete Appl. Math. (2015). doi:10.1016/j.dam.2015.10.017

  6. Brightwell, G., Heuvel, J., Stougie, L.: A linear bound on the diameter of the transportation polytope. Combinatorica 26, 133–139 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dantzig, G.: Linear Programming and Extensions. Princeton Univ Press, Princeton (1963)

    Book  MATH  Google Scholar 

  8. De Loera, J.A.: New Insights into the complexity and geometry of linear optimization. Optima Newsl. Math. Program. Soc. 87, 1–13 (2011)

    Google Scholar 

  9. De Loera J.A., Kim, E.D.: Combinatorics and geometry of transportation polytopes: an update. In: Discrete Geometry and Algebraic Combinatorics, Volume 625 of Contemporary Mathematics, pp. 37–76. American Math. Society (2014)

  10. De Loera, J.A., Kim, E.D., Onn, S., Santos, F.: Graphs of transportation polytopes. J. Comb. Theory Ser. A 116, 1306–1325 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Del Pia, A., Michini, C.: On the diameter of lattice polytopes. Discrete Comput. Geom. 55, 681–687 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Deza, A., Manoussakis, G. Onn, S.: Euler polytopes and convex matroid optimization. AdvOL-Report no. 2015/15 (2015)

  13. Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton Univ Press, Princeton (1962)

    MATH  Google Scholar 

  14. Fritzsche, K., Holt, F.B.: More polytopes meeting the conjectured Hirsch bound. Discrete Math. 205(1–3), 77–84 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Greenberg, H.J.: Diagnosing infeasibility in min-cost network flow problems part I: dual infeasibility. IMA J. Math. Manag. 1, 99–109 (1987)

    Article  MATH  Google Scholar 

  16. Holt, F.B., Klee, V.: Many polytopes meeting the conjectured Hirsch bound. Discrete Comput. Geom. 20, 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kim, E.D., Santos, F.: An update on the Hirsch conjecture. Jahresbericht der Deutschen Mathematiker-Vereinigung 112(2), 73–98 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Klee, V., Walkup, D.W.: The \(d\)-step conjecture for polyhedra of dimension \(d <\) 6. Acta Math. 117, 53–78 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  19. Klee, V., Witzgall, C.: Facets and vertices of transportation polyhedra. Mathematics of the decision science, part 1. In: Lectures in Applied Mathematics, vol. 11, pp. 257–282 (1968)

  20. Naddef, D.: The Hirsch conjecture is true for (0,1)-polytopes. Math. Program. 45, 109–110 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Orlin, J.B.: A polynomial time primal network simplex algorithm for minimum cost flows. Math. Program. 78(2), 109–129 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pak, I.: Four questions on Birkhoff polytope. Ann. Comb. 4(1), 83–90 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Santos, F.: A counterexample to the Hirsch conjecture. Ann. Math. 176, 383–412 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schrijver, A.: Combinatorial Optimization—Polyhedra and Efficiency. Springer, Berlin (2003)

    MATH  Google Scholar 

  25. Vanderbei, R.J.: Linear Programming: Foundations and Extensions. International Series in Operations Research and Management Science. Kluwer Academic, Boston (2001)

    Book  MATH  Google Scholar 

  26. Yemelichev, V.A., Kovalëv, M.M., Kravtsov, M.K.: Polytopes, Graphs and Optimisation. Cambridge University Press, Cambridge (1984)

    Google Scholar 

  27. Zadeh, N.: A bad network problem for the simplex method and other minimum cost flow algorithms. Math. Program. 5(1), 255–266 (1973)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Colorado, Denver, CO, USA

    S. Borgwardt

  2. University of California, Davis, CA, USA

    J. A. De Loera

  3. Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM, Kaiserslautern, Germany

    E. Finhold

Authors
  1. S. Borgwardt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. De Loera
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Finhold
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Borgwardt.

Additional information

S. Borgwardt gratefully acknowledges support from the Alexander-von-Humboldt Foundation. J. A. De Loera is grateful for the support received through NSF Grant DMS-1522158. J. A. De Loera and E. Finhold gratefully acknowledge the support from the Hausdorff Research Institute for Mathematics (HIM) in Bonn.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borgwardt, S., De Loera, J.A. & Finhold, E. The diameters of network-flow polytopes satisfy the Hirsch conjecture. Math. Program. 171, 283–309 (2018). https://doi.org/10.1007/s10107-017-1176-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-017-1176-x

Keywords

Mathematics Subject Classification

Search

Navigation