Abstract
We show that for every \(A\subseteq \{0,1\}^n\), there exists a polytope \(P\subseteq \mathbb {R}^n\) with \(P \cap \{0,1\}^n = A\) and extension complexity \(O(2^{n/2})\), and that there exists an \(A\subseteq \{0,1\}^n\) such that the extension complexity of any P with \(P\cap \{0,1\}^n = A\) must be at least \(2^{{n}(1-o(1))/3}\). We also remark that the extension complexity of any 0/1-polytope in \(\mathbb {R}^n\) is at most \(O(2^n/n)\) and pose the problem whether the upper bound can be improved to \(O(2^{cn})\), for \(c<1\).
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Notes
In fact, Jeroslow obtains the tight estimate \(2^{n-1}\) in Proposition 3.1 while allowing the separating polyhedron to be unbounded.
The first part of the statement can also be found in [4].
This could also be concluded from Remark 4.4.
We also have no assumption on p since the number of sign patterns can be trivially bounded by \(3^p\).
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Acknowledgements
We are grateful to Fedor Part for discussions, Gennadiy Averkov and Emil Jeřábek for useful references. This work was done while Talebanfard was participating in the program Satisfiability: Theory, Practice, and Beyond at the Simons Institute for the Theory of Computing.
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Hrubeš, P., Talebanfard, N. On the Extension Complexity of Polytopes Separating Subsets of the Boolean Cube. Discrete Comput Geom 70, 268–278 (2023). https://doi.org/10.1007/s00454-022-00419-3
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DOI: https://doi.org/10.1007/s00454-022-00419-3