Some 0/1 polytopes need exponential size extended formulations
Author(s)
Rothvoss, Thomas
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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.
Date issued
2012-07Department
Massachusetts Institute of Technology. Department of MathematicsJournal
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
ISSN
0025-5610
1436-4646