Two-Level Polytopes with a Prescribed Facet

Fiorini, Samuel; Fisikopoulos, Vissarion; Macchia, Marco

doi:10.1007/978-3-319-45587-7_25

Cited by 6 publications

(11 citation statements)

References 16 publications

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“…holds. To this end, consider the linear subspace Π ⊆ U spanned by v and W and let σ : U → Π denote the orthogonal projection on Π. by (8). Moreover, recall that π(B) does not contain opposite points by Claim 1 (iii).…”

Section: Claim 2 Every Point In π(B) Has At Most Two Preimages In Bmentioning

confidence: 99%

“…Details on these connections and several recent studies on 2-level polytopes can be found in [2,3,4,5,8,13,14,15]. Among them are extensive experimental studies by Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich [4,8,5], which led to a beautiful conjecture about the combinatorial structure of 2-level polytopes. It is easy to see that for a d-dimensional 2-level polytope P the number of vertices f 0 (P ) and the number of facets f d−1 (P ) are both bounded by 2 d .…”

Section: Introductionmentioning

confidence: 96%

“…γ b ′′ } by(8). Thus, again by the maximality of B, (a scaling of) the vector b 2 is contained in B, and since b 2 is orthogonal to A 0 and a * ∈ A 1 , we have ϕ(b 2 ) ≥ dim(A 0 ) + 1.…”

mentioning

confidence: 91%

“…In contrast, it is known that, among all d-dimensional polytopes, 2-level polytopes admit smallest possible semidefinite extended formulations [13]. Details on these connections and several recent studies on 2-level polytopes can be found in [2,3,4,5,8,13,14,15]. Among them are extensive experimental studies by Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich [4,8,5], which led to a beautiful conjecture about the combinatorial structure of 2-level polytopes.…”

Section: Introductionmentioning

confidence: 99%

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Binary scalar products

Kupavskii,

Weltge

2020

Preprint

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Let A, B ⊆ R d both span R d such that a, b ∈ {0, 1} holds for all a ∈ A, b ∈ B. We show that |A| • |B| ≤ (d + 1)2 d . This allows us to settle a conjecture by Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich ( 2015) concerning 2-level polytopes. Such polytopes have the property that for every facet-defining hyperplane H there is a parallel hyperplane H ′ such that H ∪ H ′ contain all vertices. The authors conjectured that for every d-dimensional 2-level polytope P the product of the number of vertices of P and the number of facets of P is at most d2 d+1 , which we show to be true.

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Section: Claim 2 Every Point In π(B) Has At Most Two Preimages In Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

mentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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Binary scalar products

Kupavskii,

Weltge

2020

Preprint

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“…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.…”

mentioning

confidence: 99%

On 2-Level Polytopes Arising in Combinatorial Settings

Aprile¹,

Cevallos²,

Faenza³

2018

SIAM J. Discrete Math.

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2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level polytopes arising in combinatorial settings. Our first contribution is proving that f0(P )f d−1 (P ) ≤ d2 d+1 for a large collection of families of such polytopes P . Here f0(P ) (resp. f d−1 (P )) is the number of vertices (resp. 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 leads to a number of results that we believe to be of independent interest: a trade-off formula for the number of cliques and stable sets in a graph; a description of stable matching polytopes as affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree. IntroductionLet P ⊆ R d be a polytope. We say that P is 2-level if, for each facet F of P , all the vertices of P that are not vertices of F lie in the same translate of the affine hull of F . Equivalently, P is 2-level if and only if it has theta-rank 1 [20], or all its pulling triangulations are unimodular [50], or it has a slack matrix with entries in {0, 1} [7]. Those last three definitions appeared in papers from the semidefinite programming, statistics, and polyhedral combinatorics communities respectively, showing that 2-level polytopes naturally arise in many areas of mathematics. 2-level polytopes generalize Birkhoff [53], Hanner [28], and Hansen polytopes [29], order polytopes and chain polytopes of posets [49], spanning tree polytopes of series-parallel graphs [24], stable matching polytopes [27], some min up/down polytopes [37], and stable set polytopes of perfect graphs [10].

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Enumeration of 2-level polytopes

et al. 2018

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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.

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Two-Level Polytopes with a Prescribed Facet

Cited by 6 publications

References 16 publications

Binary scalar products

Binary scalar products

On 2-Level Polytopes Arising in Combinatorial Settings

Enumeration of 2-level polytopes

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