article

Extended Formulations for Polygons

Authors:

Samuel Fiorini,

Thomas Rothvoβ,

Hans Raj TiwaryAuthors Info & Claims

Discrete & Computational Geometry, Volume 48, Issue 3

Pages 658 - 668

Published: 01 October 2012 Publication History

Abstract

The extension complexity of a polytope P is the smallest integer k such that P is the projection of a polytope Q with k facets. We study the extension complexity of n-gons in the plane. First, we give a new proof that the extension complexity of regular n-gons is O(logn), a result originating from work by Ben-Tal and Nemirovski (Math. Oper. Res. 26(2), 193---205, 2001). Our proof easily generalizes to other permutahedra and simplifies proofs of recent results by Goemans (2009), and Kaibel and Pashkovich (2011). Second, we prove a lower bound of $\sqrt{2n}$ on the extension complexity of generic n-gons. Finally, we prove that there exist n-gons whose vertices lie on an O(n) O(n2) integer grid with extension complexity $\varOmega (\sqrt{n}/\sqrt{\log n})$.

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cover image Discrete & Computational Geometry

Discrete & Computational Geometry Volume 48, Issue 3

October 2012

294 pages

ISSN:0179-5376

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Copyright © Copyright © 2012 Springer Science+Business Media, LLC.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 October 2012

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Cited By

Shitov Y(2024)Further -Complete Problems with PSD Matrix FactorizationsFoundations of Computational Mathematics10.1007/s10208-023-09610-124:4(1225-1248)Online publication date: 1-Aug-2024
https://dl.acm.org/doi/10.1007/s10208-023-09610-1
Shitov Y(2019)Euclidean Distance Matrices and Separations in Communication Complexity TheoryDiscrete & Computational Geometry10.1007/s00454-018-9988-x61:3(653-660)Online publication date: 1-Apr-2019
https://dl.acm.org/doi/10.1007/s00454-018-9988-x
Vandaele AGlineur FGillis N(2018)Algorithms for positive semidefinite factorizationComputational Optimization and Applications10.1007/s10589-018-9998-x71:1(193-219)Online publication date: 1-Sep-2018
https://dl.acm.org/doi/10.1007/s10589-018-9998-x
Grande FPadrol ASanyal R(2018)Extension Complexity and Realization Spaces of HypersimplicesDiscrete & Computational Geometry10.1007/s00454-017-9925-459:3(621-642)Online publication date: 1-Apr-2018
https://dl.acm.org/doi/10.1007/s00454-017-9925-4
Fawzi HSaunderson JParrilo P(2017)Equivariant Semidefinite Lifts of Regular PolygonsMathematics of Operations Research10.1287/moor.2016.081342:2(472-494)Online publication date: 1-May-2017
https://dl.acm.org/doi/10.1287/moor.2016.0813
Rothvoss T(2017)The Matching Polytope has Exponential Extension ComplexityJournal of the ACM10.1145/312749764:6(1-19)Online publication date: 28-Sep-2017
https://dl.acm.org/doi/10.1145/3127497
Vandaele AGillis NGlineur FTuyttens D(2016)Heuristics for exact nonnegative matrix factorizationJournal of Global Optimization10.1007/s10898-015-0350-z65:2(369-400)Online publication date: 1-Jun-2016
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Braun GPokutta S(2016)Common Information and Unique DisjointnessAlgorithmica10.1007/s00453-016-0132-076:3(597-629)Online publication date: 1-Nov-2016
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Braun GPokutta SIndyk P(2015)The matching polytope does not admit fully-polynomial size relaxation schemesProceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms10.5555/2722129.2722186(837-846)Online publication date: 4-Jan-2015
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Braun GPokutta S(2015)The Matching Problem Has No Fully Polynomial Size Linear Programming Relaxation SchemesIEEE Transactions on Information Theory10.1109/TIT.2015.246586461:10(5754-5764)Online publication date: 1-Oct-2015
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