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., \(\{y \in \mathbb {R}^d \mid x+\lambda y \in C,\,\forall x\in C,\,\forall \lambda \ge 0\}\).
References
Aprile, M., Cevallos, A., Faenza, Y.: On 2-level polytopes arising in combinatorial settings. SIAM J. Discrete Math. 32(3), 1857–1886 (2018)
Birkhoff, G.: Tres observaciones sobre el algebra lineal. Univ. Nac. Tucumán Rev. Ser. A 5, 147–151 (1946)
Bohn, A., Faenza, Yu., Fiorini, S., Fisikopoulos, V., Macchia, M., Pashkovich, K.: Enumeration of 2-level polytopes (2017). arXiv:1703.01943
Chvátal, V.: On certain polytopes associated with graphs. J. Comb. Theory Ser. B 18, 138–154 (1975)
Deza, M.M., Laurent, M.: Geometry of Cuts and Metrics. Algorithms and Combinatorics. Springer, Berlin (2009)
Erdős, P., Kleitman, D.J., Rothschild, B.L.: Asymptotic enumeration of \(K_n\)-free graphs. In: Colloquio Internazionale sulle Teorie Combinatorie, vol. II. Atti dei Convegni Lincei, vol. 17, pp. 19–27. Accademia Nazionale dei Lincei, Rome (1976)
Gouveia, J., Grappe, R., Kaibel, V., Pashkovich, K., Robinson, R.Z., Thomas, R.R.: Which nonnegative matrices are slack matrices? Linear Algebra Appl. 439(10), 2921–2933 (2013)
Gouveia, J., Parrilo, P.A., Thomas, R.R.: Theta bodies for polynomial ideals. SIAM J. Optim. 20(4), 2097–2118 (2010)
Gouveia, J., Pashkovich, K., Robinson, R.Z., Thomas, R.R.: Four-dimensional polytopes of minimum positive semidefinite rank. J. Comb. Theory Ser. A 145, 184–226 (2017)
Gouveia, J., Robinson, R.Z., Thomas, R.R.: Polytopes of minimum positive semidefinite rank. Discrete Comput. Geom. 50(3), 679–699 (2013)
Grande, F.: On \(k\)-Level Matroids: Geometry and Combinatorics. PhD thesis, Freie Universität Berlin (2015)
Grande, F., Rué, J.: Many 2-level polytopes from matroids. Discrete Comput. Geom. 54(4), 954–979 (2015)
Hanner, O.: Intersections of translates of convex bodies. Math. Scand. 4, 65–87 (1956)
Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)
Lee, T., Shraibman, A.: Lower Bounds in Communication Complexity. Now Publishers, Hanover (2009)
Lovász, L., Saks, M.: Communication complexity and combinatorial lattice theory. J. Comput. Syst. Sci. 47(2), 322–349 (1993)
Macchia, M.: Two Level Polytopes: Geometry and Optimization. PhD thesis, Université Libre de Bruxelles (2018)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. Wiley, Chichester (1986)
Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1(1), 9–23 (1986)
Sullivant, S.: Compressed polytopes and statistical disclosure limitation. Tohoku Math. J. 58(3), 433–445 (2006)
Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)
Yao, A.C.: Some complexity questions related to distributed computing. In: Proceedings of the 11th Annual ACM Symposium on Theory of Computing (STOC’79), pp. 209–213. ACM, New York (1979)
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