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An Algebraic Approach to Symmetric Extended Formulations

  • Gábor Braun
  • Sebastian Pokutta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7422)

Abstract

Extended formulations are an important tool to obtain small (even compact) formulations of polytopes by representing them as projections of higher dimensional ones. It is an important question whether a polytope admits a small extended formulation, i.e., one involving only a polynomial number of inequalities in its dimension. For the case of symmetric extended formulations (i.e., preserving the symmetries of the polytope) Yannakakis established a powerful technique to derive lower bounds and rule out small formulations. We rephrase the technique of Yannakakis in a group-theoretic framework. This provides a different perspective on symmetric extensions and considerably simplifies several lower bound constructions.

Keywords

symmetric extended formulations polyhedral combinatorics group theory representation theory matching polytope 

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References

  1. Conforti, M., Cornuéjols, G., Zambelli, G.: Extended formulations in combinatorial optimization. 4OR: A Quarterly Journal of Operations Research 8(1), 1–48 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. Dixon, J.D., Mortimer, B.: Permutation groups. Springer (1996) ISBN 0387945997Google Scholar
  3. Faenza, Y., Kaibel, V.: Extended formulations for packing and partitioning orbitopes. Mathematics of Operations Research 34(3), 686–697 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.: Combinatorial Bounds on Nonnegative Rank and Extended Formulations. Arxiv preprint arXiv:1111.0444 (2011)Google Scholar
  5. Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds. Arxiv preprint arxiv:1111.0837 (2011)Google Scholar
  6. Goemans, M.X.: Smallest compact formulation for the permutahedron, preprint (2009)Google Scholar
  7. Kaibel, V.: Extended formulations in combinatorial optimization. Arxiv preprint arXiv:1104.1023 (2011)Google Scholar
  8. Kaibel, V., Pashkovich, K.: Constructing Extended Formulations from Reflection Relations. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 287–300. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. Kaibel, V., Pashkovich, K., Theis, D.O.: Symmetry Matters for the Sizes of Extended Formulations. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 135–148. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. Köppe, M., Louveaux, Q., Weismantel, R.: Intermediate integer programming representations using value disjunctions. Discrete Optimization 5(2), 293–313 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  11. Pashkovich, K.: Symmetry in Extended Formulations of the Permutahedron. Arxiv preprint arXiv:0912.3446 (2009)Google Scholar
  12. Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences 43(3), 441–466 (1991)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Gábor Braun
    • 1
  • Sebastian Pokutta
    • 2
  1. 1.Institut für InformatikUniversität LeipzigLeipzigGermany
  2. 2.Department of MathematicsFriedrich-Alexander-University of Erlangen-NürnbergErlangenGermany

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