Bounds on the number of 2-level polytopes, cones and configurations

TL;DR

This paper establishes an exponential upper bound on the number of affine and linear equivalence classes of 2-level polytopes, cones, and configurations in d dimensions, confirming a conjecture and providing bounds through geometric and combinatorial methods.

Contribution

It proves a tight upper bound on the number of 2-level polytopes, cones, and configurations, and relates these classes to faces of the correlation cone, confirming a conjecture by Bohn et al.

Findings

Upper bound of 2^{O(d^2 polylog d)} on the number of classes

Lower bound of 2^{Ω(d^2)} from stable set polytopes

Positive resolution of Bohn et al.'s conjecture

Abstract

We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ 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 $2^{\Omega(d^2)}$ lower bound, by estimating the number of nonequivalent stable set polytopes of bipartite graphs.