Skip to main content
SpringerLink
Log in

Lower Bounds on the Sizes of Integer Programs without Additional Variables

  • Conference paper
Integer Programming and Combinatorial Optimization (IPCO 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8494))

Abstract

For a given set X ⊆ ℤd of integer points, we investigate the smallest number of facets of any polyhedron whose set of integer points is conv(X) ∩ ℤd . This quantity, which we call the relaxation complexity of X , corresponds to the smallest number of linear inequalities of any integer program having X as the set of feasible solutions that does not use auxiliary variables. We show that the use of auxiliary variables is essential for constructing polynomial size integer programming formulations in many relevant cases. In particular, we provide asymptotically tight exponential lower bounds on the relaxation complexity of the integer points of several well-known combinatorial polytopes, including the traveling salesman polytope and the spanning tree polytope.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In: STOC, pp. 95–106 (2012)

    Google Scholar 

  2. Gavish, B., Graves, S.C.: The travelling salesman problem and related problems. Technical report, Operations Research Center, Massachusetts Institute of Technology (1978)

    Google Scholar 

  3. Goemans, M.: Smallest compact formulation for the permutahedron. Kaibel, V., Thomas, R. (eds.) To appear in a forthcoming issue of Mathematical Programming, Series B “Lifts of Convex Sets” (2014)

    Google Scholar 

  4. Jeroslow, R.: On defining sets of vertices of the hypercube by linear inequalities. Discrete Mathematics 11(2), 119–124 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  5. Lee, J., Margot, F.: On a binary-encoded ilp coloring formulation. INFORMS Journal on Computing 19(3), 406–415 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kipp Martin, R.: Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Miller, C.E., Tucker, A.W., Zemlin, R.A.: Integer programming formulation of traveling salesman problems. J. ACM 7(4), 326–329 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  8. Minkowski, H.: Geometrie der Zahlen. Teubner Verlag, Leipzig (1896)

    Google Scholar 

  9. Rado, R.: An inequality. J. London Math. Soc. 27, 1–6 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  10. Schrijver, A.: Combinatorial Optimization – Polyhedra and Efficiency. Springer (2003)

    Google Scholar 

  11. Schrijver, A.: Theory of linear and integer programming. John Wiley & Sons, Inc., New York (1986)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Otto-von-Guericke-Universität Magdeburg, Germany

    Volker Kaibel & Stefan Weltge

Authors
  1. Volker Kaibel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Weltge
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Industrial and Operations Engineering, University of Michigan, Beal Avenue, 1205, Ann Arbor, MI, USA

    Jon Lee

  2. Research Institute for Discrete Mathematics, University of Bonn, Lennéstr. 2, P.O. Box, 53113, Bonn, Germany

    Jens Vygen

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Kaibel, V., Weltge, S. (2014). Lower Bounds on the Sizes of Integer Programs without Additional Variables. In: Lee, J., Vygen, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2014. Lecture Notes in Computer Science, vol 8494. Springer, Cham. https://doi.org/10.1007/978-3-319-07557-0_27

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-07557-0_27

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-07556-3

  • Online ISBN: 978-3-319-07557-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation