Enumeration of 2-level polytopes

Abstract

A (convex) polytope P is said to be 2-level if for every direction of hyperplanes which is facet-defining for P , the vertices of P can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for d 7. Our approach is inductive: for each fixed (d − 1)-dimensional 2-level polytope P 0 , we enumerate all d-dimensional 2-level polytopes P that have P 0 as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet P 0 , we obtain all 2-level polytopes in dimension d.(1,3,4,5) Université libre de Bruxelles, Brussels, Belgium Lemma 4. Let P be a d-polytope having facet-defining inequalities g 1 (x) 0, . . . , g m (x) 0, and vertices v 1 , . . . , v n . If σ denotes a map from the affine hull aff(P ) of P to R m defined by σ(x) i := g i (x) for all x ∈ aff(P ), then the polytopes P and σ(P ) are affinely equivalent.Proof. The map σ : aff(P ) → R m is affine, and injective because it maps the vertices of any simplicial core for P to affinely independent points. The result follows.By definition, a polytope P is 2-level if and only if S(P ) can be scaled to be a 0/1 matrix. Given a 2-level polytope, we henceforth always assume that its facet-defining inequalities are scaled so that the slacks are 0/1. Thus, the slack embedding of a 2-level polytope depends only on the support 1 of its slack matrix, which only depends on its combinatorial structure. As a consequence, we have the following result:Lemma 5. Two 2-level polytopes are affinely equivalent if and only if they have the same combinatorial type.

Slack matrices, $k$-products, and $2$-level polytopes

“…Moreover, if S is a slack matrix of polytopes P 1 , P 2 , then P 1 and P 2 are affinely isomorphic. For proofs of these facts and more properties of slack matrices, see, e.g., [8].…”

On 2-Level Polytopes Arising in Combinatorial Settings

“…We consider possible generalizations of the conjecture based on the H−embedding of 2-level polytopes. Those are defined in [7], where it is shown that the family of 2-level polytopes of dimension d is affinely equivalent to the family of integral polytopes of the form…”

“…facets) of P , and d is its dimension. Whether this holds for all 2-level polytopes was asked in [7], and experimental results from [16] showed it true for d ≤ 7. The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structures.…”

“…This makes 2-level polytopes quite different from "random" 0/1 polytopes, that have (d/ log d) Θ(d) facets [5]. Experimental results from [7,16] suggest that this separation could be even stronger: up to d = 7, the product of the number of facets f d−1 (P ) and the number of vertices f 0 (P ) of a d-dimensional 2-level polytope P does not exceed d2 d+1 . In [7], it is asked whether this always holds, and in their journal version the question is turned into a conjecture.…”

An Algebraic Approach to Projective Uniqueness with an Application to Order Polytopes

“…Such polytopes play an important role in the theory of semidefinite representations of polyhedra and have been the focus of recent interest. Moreover, they comprise a very large family that includes many interesting polytopes; see for example [ACF18] for combinatorially relevant examples and [BFFFMP19] for a full enumeration in dimension up to 7.…”