Abstract
We prove an upper bound of the form \(2^{O(d^2 \mathop {\mathrm {polylog}}d)}\) on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones, and d-configurations. This in particular answers positively a conjecture of Bohn et al. on 2-level polytopes. We obtain our upper bound by relating affine (resp. linear) equivalence classes of 2-level d-polytopes, d-cones, and d-configurations to faces of the correlation cone. We complement this with a \(2^{\varOmega (d^2)}\) lower bound, by estimating the number of nonequivalent stable set polytopes of bipartite graphs.
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Notes
A matrix M is said to be totally unimodular provided that the determinant of every square submatrix of M is either 0, 1, or \(-1\), see for instance [18].
Throughout, we assume that systems of linear inequalities do not have repeated inequalities.
Given a nonempty convex set \(C \subseteq \mathbb {R}^d\), the recession cone of C is the set of all directions along which we can move indefinitely and still be in C, i.e., \(\{y \in \mathbb {R}^d \mid x+\lambda y \in C,\,\forall x\in C,\,\forall \lambda \ge 0\}\).
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Acknowledgements
This work was done while the authors were visiting the Simons Institute for the Theory of Computing. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF Grant # CCF-1740425.
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We acknowledge the support from ERC Grant FOREFRONT (Grant Agreement No. 615640) funded by the European Research Council under the EU’s 7th Framework Programme (FP7/2007-2013).
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Fiorini, S., Macchia, M. & Pashkovich, K. Bounds on the Number of 2-Level Polytopes, Cones, and Configurations. Discrete Comput Geom 65, 587–600 (2021). https://doi.org/10.1007/s00454-020-00181-4
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DOI: https://doi.org/10.1007/s00454-020-00181-4