Simple 0/1-Polytopes

Kaibel, Volker; Wolff, Martin

doi:10.1006/eujc.1999.0328

Cited by 17 publications

(13 citation statements)

References 6 publications

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“…The upper bounds on g nfac (d, n) provided by the polytopes P (d, n) in Proposition 8 are not sharp, at least not for all parameters d and n. This follows, for instance, from the examples of Cartesian products of r 0/1-simplices of dimension d 1 ,…,d r (which are precisely the simple 0/1-polytopes, see Kaibel and Wolff [8]). Such a product is a 0/1-polytope of dimension d = d i with (d i + 1) vertices and d + r facets.…”

Section: An Upper Bound On the Minimal Number Of Facetsmentioning

confidence: 95%

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Revlex-initial 0/1-polytopes

Gillmann

Kaibel

2006

Journal of Combinatorial Theory, Series A

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We introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In particular, we use these polytopes in order to prove that the minimum numbers g nfac (d, n) of facets and the minimum average degree g avdeg (d, n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy g nfac (d, n) 3d and g avdeg (d, n) d + 4. We furthermore show that, despite the sparsity of their graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at least one.

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Section: An Upper Bound On the Minimal Number Of Facetsmentioning

confidence: 95%

“…For instance, a 0/1-polytope is simple if and only if it is the product of 0/1-simplices [8]. Thus, d-dimensional simple 0/1-polytopes with n vertices do only exist if there is a factorization n = n i of n with d = (n i − 1).…”

Section: Introductionmentioning

confidence: 97%

Revlex-initial 0/1-polytopes

Gillmann

Kaibel

2006

Journal of Combinatorial Theory, Series A

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“…P : f-vector (8,21,22,9) Q 1 : f-vector (7,19,23,11) Q 2 : f-vector (8,25,32,15) Let us now consider the general case of a d-dimensional face of a permutation polytope. d = 2: The triangle and the square are the only two-dimensional 0/1-polytopes.…”

Section: Classification Of 4-dimensional Facesmentioning

confidence: 99%

On permutation polytopes

Baumeister

Haase

Nill

et al. 2009

Advances in Mathematics

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In this paper we lay the foundations for the study of permutation polytopes: the convex hull of a group of permutation matrices.We clarify the relevant notions of equivalence, prove a product theorem, and discuss centrally symmetric permutation polytopes. We provide a number of combinatorial properties of (faces of) permutation polytopes. As an application, we classify 4-dimensional permutation polytopes and the corresponding permutation groups. Classification results and further examples are made available online.We conclude with several questions suggested by a general finiteness result.

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“…It is known that Bøgvad's conjecture holds for compressed polytopes. Indeed, a compressed polytope is integral-affinely equivalent to a 0-1 polytope by Proposition 5.1, and a smooth 0-1 polytope is the product of 0-1 simplices by [23], hence the toric ideal of a smooth compressed polytope is generated in degree two (say by Lemma 4.2), moreover it possesses a quadratic quadratic Gröbner basis (by [37,Theorem 13]). Note that when (Q, θ) is tight (cf.…”

Section: Toric Ideals Of Compressed Polytopesmentioning

confidence: 99%

On the equations and classification of toric quiver varieties

Domokos

Joó

2016

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight there is an associated quasiprojective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3. In each fixed dimension d up to isomorphism there are only finitely many ddimensional toric quiver varieties. A procedure for their classification is outlined.

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Simple 0/1-Polytopes

Cited by 17 publications

References 6 publications

Revlex-initial 0/1-polytopes

Revlex-initial 0/1-polytopes

On permutation polytopes

On the equations and classification of toric quiver varieties

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