Some 0/1 polytopes need exponential size extended formulations
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10107_2012_Article_574.pdf
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Author(s)
Rothvoss, Thomas
Date Issued
July 2012
Journal
Mathematical Programming
Publisher
Springer Berlin Heidelberg
Citation
Rothvoß, Thomas. “Some 0/1 Polytopes Need Exponential Size Extended Formulations.” Mathematical Programming 142.1–2 (2013): 255–268.
Version
Author's final manuscript
Abstract
We prove that there are 0/1 polytopes P⊆R[superscript n] that do not admit a compact LP formulation. More precisely we show that for every n there is a set X⊆{0,1}[superscript n] such that conv(X) must have extension complexity at least 2[superscript n/2⋅(1−o(1)] . In other words, every polyhedron Q that can be linearly projected on conv(X) must have exponentially many facets. In fact, the same result also applies if conv(X) is restricted to be a matroid polytope. Conditioning on NP⊈P[subscript /poly], our result rules out the existence of a compact formulation for any NP -hard optimization problem even if the formulation may contain arbitrary real numbers.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s10107-012-0574-3
