Skip to main content
SpringerLink
Log in

Quadratic diameter bounds for dual network flow polyhedra

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

Abstract

Both the combinatorial and the circuit diameter of polyhedra are of interest to the theory of linear programming for their intimate connection to a best-case performance of linear programming algorithms. We study the diameters of dual network flow polyhedra associated to b-flows on directed graphs \(G=(V,E)\) and prove quadratic upper bounds for both of them: the minimum of \((|V|-1)\cdot |E|\) and \(\frac{1}{6}|V|^3\) for the combinatorial diameter, and \(\frac{|V|\cdot (|V|-1)}{2}\) for the circuit diameter. Previously, bounds on these diameters have only been known for bipartite graphs. The situation is much more involved for general graphs. In particular, we construct a family of dual network flow polyhedra with members that violate the circuit diameter bound for bipartite graphs by an arbitrary additive constant.

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Borgwardt, S., Finhold, E., Hemmecke, R.: On the circuit diameter of dual transportation polyhedra. SIAM J. Discret. Math. 29(1), 113–121 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dantzig, G.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)

    MATH  Google Scholar 

  4. De Loera, J.A., Hemmecke, R., Köppe, M.: Algebraic and Geometric Ideas in the Theory of Discrete Optimization, volume 14 of MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2013)

  5. De Loera, J.A., Hemmecke, R., Lee, J.: Augmentation Algorithms for Linear and Integer-Linear Programming: From Edmonds–Karp to Bland and Beyond. eprint arXiv:1408.3518 (2014)

  6. De Loera, J.A., Kim, E.D.: Combinatorics and Geometry of Transportation Polytopes: An Update. Discrete Geometry and Algebraic Combinatorics, Volume 625 of Contemporary Mathematics, pp. 37–76 (2014)

  7. Finhold, E.: Circuit Diameters and Their Application to Transportation Problems. PhD thesis, Technische Universität München (2015)

  8. Graver, J.E.: On the foundation of linear and integer programming I. Math. Program. 9, 207–226 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

  10. Klee, V., Walkup, D.W.: The \(d\)-step conjecture for polyhedra of dimension \(d < 6\). Acta Math. 133, 53–78 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Santos, F.: A counterexample to the Hirsch conjecture. Ann. Math. 176(1), 383–412 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Stephen, T., Yusun, T.: Circuit Diameter and Klee-Walkup Constructions. eprint arXiv:1503.05252 (2015)

Download references

Acknowledgments

The second author gratefully acknowledges the support from the graduate program TopMath of the Elite Network of Bavaria and the TopMath Graduate Center of TUM Graduate School at Technische Universität München.

Author information

Authors and Affiliations

  1. Technische Universität München, Munich, Germany

    Steffen Borgwardt, Elisabeth Finhold & Raymond Hemmecke

Authors
  1. Steffen Borgwardt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elisabeth Finhold
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Raymond Hemmecke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Steffen Borgwardt.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borgwardt, S., Finhold, E. & Hemmecke, R. Quadratic diameter bounds for dual network flow polyhedra. Math. Program. 159, 237–251 (2016). https://doi.org/10.1007/s10107-015-0956-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-015-0956-4

Keywords

Mathematics Subject Classification

Search

Navigation