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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Conforti, M., Cornuéjols, G., Zambelli, G.: Extended formulations in combinatorial optimization. 4OR: A Quarterly Journal of Operations Research 8(1), 1–48 (2010)
Dixon, J.D., Mortimer, B.: Permutation groups. Springer (1996) ISBN 0387945997
Faenza, Y., Kaibel, V.: Extended formulations for packing and partitioning orbitopes. Mathematics of Operations Research 34(3), 686–697 (2009)
Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.: Combinatorial Bounds on Nonnegative Rank and Extended Formulations. Arxiv preprint arXiv:1111.0444 (2011)
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)
Goemans, M.X.: Smallest compact formulation for the permutahedron, preprint (2009)
Kaibel, V.: Extended formulations in combinatorial optimization. Arxiv preprint arXiv:1104.1023 (2011)
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)
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)
Köppe, M., Louveaux, Q., Weismantel, R.: Intermediate integer programming representations using value disjunctions. Discrete Optimization 5(2), 293–313 (2008)
Pashkovich, K.: Symmetry in Extended Formulations of the Permutahedron. Arxiv preprint arXiv:0912.3446 (2009)
Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences 43(3), 441–466 (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Braun, G., Pokutta, S. (2012). An Algebraic Approach to Symmetric Extended Formulations. In: Mahjoub, A.R., Markakis, V., Milis, I., Paschos, V.T. (eds) Combinatorial Optimization. ISCO 2012. Lecture Notes in Computer Science, vol 7422. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32147-4_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-32147-4_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32146-7
Online ISBN: 978-3-642-32147-4
eBook Packages: Computer ScienceComputer Science (R0)