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Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube
[chapter]
1999
Lecture Notes in Computer Science
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Finally, we provide a family of polytopes contained in the 0/1-cube whose Chvátal rank is at least (1 + )n, for some > 0. ...
Moreover, we also demonstrate that the rank of any polytope in the 0/1-cube whose integer hull is defined by inequalities with constant coefficients is O(n). ...
The authors are grateful to Alexander Bockmayr, Volker Priebe, and Günter Ziegler for helpful comments on an earlier ver- ...
doi:10.1007/3-540-48777-8_11
fatcat:x3m6vjusmrbpdokawg4xsvdtye
Characterizing Polytopes Contained in the 0/1-Cube with Bounded Chvátal-Gomory Rank
[article]
2017
arXiv
pre-print
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We prove that R has bounded Chvátal-Gomory rank (CG-rank) provided that S has bounded notch and bounded gap, where the notch is the minimum integer p such that all p-dimensional faces of the 0/1-cube have ...
Let S ⊆{0,1}^n and R be any polytope contained in [0,1]^n with R ∩{0,1}^n = S. ...
In this paper, we give new bounds on the CG-rank of a polytope R contained in [0, 1] n that only depend on properties of S = R∩ {0, 1} n and not on R itself. ...
arXiv:1611.06593v3
fatcat:vmmlmfbnwzd2do2vuboy6r32xq
Lower bounds for the Chvátal–Gomory rank in the 0/1 cube
2011
Operations Research Letters
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We provide new families of polytopes in the 0/1 cube with high rank and we describe a deterministic family achieving a rank of at least (1 + 1/e)n − 1 > n. ...
We revisit the method of Chvátal, Cook, and Hartmann to establish lower bounds on the Chvátal-Gomory rank and develop a simpler method. ...
We would also like to thank Michael Joswig for providing us with an explicit inequality description of the polytope defined in Lemma 3.1 (cf. Remark 3.3) and Santanu Dey for pointing us to [10] . ...
doi:10.1016/j.orl.2011.03.001
fatcat:jza5vnw7dzattnh7rl6ixajcvi
Integer-empty polytopes in the 0/1-cube with maximal Gomory–Chvátal rank
2011
Operations Research Letters
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In particular, we show that the first Gomory-Chvátal closure of all these polytopes is identical. ...
We provide a complete characterization of all polytopes P ⊆ [0, 1] n with empty integer hull whose Gomory-Chvátal rank is n (and, therefore, maximal). ...
ACKNOWLEDGEMENTS The authors are grateful to the reviewer for his constructive comments, which really helped to improve the presentation of this paper. ...
doi:10.1016/j.orl.2011.09.004
fatcat:henkwj6iknafnohxx3bpkzlge4
On the Chvátal rank of polytopes in the 0/1 cube
1999
Discrete Applied Mathematics
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In this paper, we study polytopes in the 0=1 cube, which are of particular interest in combinatorial optimization. ...
It is always ÿnite, but already the Chvà atal rank of polytopes in R 2 can be arbitrarily large. ...
Acknowledgements G unter Ziegler suggested an improved version of Lemma 3, which we present here. Thanks also to William Cook, Georg Struth, and G unter Ziegler for inspiring discussions. ...
doi:10.1016/s0166-218x(99)00156-0
fatcat:l5e2tmg4bng4doj4o4mbyxmjwa
Characterizing Polytopes in the 0/1-Cube with Bounded Chvátal-Gomory Rank
2018
Mathematics of Operations Research
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We prove that R has bounded Chvátal-Gomory rank (CG-rank) provided that S has bounded notch and bounded gap, where the notch is the minimum integer p such that all p-dimensional faces of the 0/1-cube have ...
Let S ⊆ {0, 1} n and R be any polytope contained in [0, 1] n with R ∩ {0, 1} n = S. ...
In this paper, we give new bounds on the CG-rank of a polytope R contained in [0, 1] n that only depend on properties of S = R∩ {0, 1} n and not on R itself. ...
doi:10.1287/moor.2017.0880
fatcat:kygmgxshpffdpkaqw2karwuj2i
Page 7114 of Mathematical Reviews Vol. , Issue 2004i
[page]
2004
Mathematical Reviews
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Third, a family of polytopes contained in the 0/1-cube is provided whose Chvatal rank is at least (1+ e)n, for some ¢ > 0. ...
0/1-cube. ...
Page 6701 of Mathematical Reviews Vol. , Issue 2000i
[page]
2000
Mathematical Reviews
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the Chyatal rank of polytopes in the 0/1 cube. ...
Chvatal and others have investigated the Chvatal rank in the last 30 years. The present paper yields new statements about the Chvatal rank of polytopes in the 0/1 cube. ...
Page 3769 of Mathematical Reviews Vol. , Issue 2000e
[page]
2000
Mathematical Reviews
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The authors prove the O(n*lgn) upper bound on the Chvatal rank of polytopes in the 0/1-cube in Section 3. ...
the 0/1-cube. ...
Random half-integral polytopes
2011
Operations Research Letters
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We show that half-integral polytopes obtained as the convex hull of a random set of half-integral points of the 0/1 cube have rank as high as Ω(log n/ log log n) with positive probability -even if the ...
size of the set relative to the total number of half-integral points of the cube tends to 0. ...
We choose a subset S n of the half-integral points of the n-dimensional 0/1-cube except the vertices. ...
doi:10.1016/j.orl.2011.03.003
fatcat:izijrux3lbd2bmik3ykydn4tli
0/1 Polytopes with Quadratic Chvátal Rank
[chapter]
2013
Lecture Notes in Computer Science
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We prove that there is a polytope contained in the 0/1 cube that has Chvátal rank Ω(n 2 ), closing the gap up to a logarithmic factor. ...
The number of iterations of this procedure that are needed until the integral hull of P is reached is called the Chvátal rank. ...
In particular, the question whether there is any superlinear lower bound on the rank of a polytope in the 0/1 cube is open since many years (see e.g. Ziegler [Zie00] ). ...
doi:10.1007/978-3-642-36694-9_30
fatcat:jfi7ly5oxzfbxokmttrbhatfhm
0/1 Polytopes with Quadratic Chvátal Rank
2017
Operations Research
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In particular, the question whether there is any superlinear lower bound on the rank of a polytope in the 0/1 cube is open since many years (see e.g. Ziegler [Zie00]). ...
We prove that there is a polytope contained in the 0/1 cube that has Chvátal rank Ω(n 2 ), closing the gap up to a logarithmic factor. ...
Acknowledgements The authors are very grateful to Michel X. Goemans for useful discussions. ...
doi:10.1287/opre.2016.1549
fatcat:atxjdchjjfe5tbn3labrdxltyu
On the Rank of Cutting-Plane Proof Systems
[chapter]
2010
Lecture Notes in Computer Science
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We exhibit a family of polytopes without integral points contained in the n-dimensional 0/1-cube that has rank Ω(n/ log n) for every proof system in our class. ...
A (tight) lower bound of n for the rank of the Gomory-Chvátal procedure for polytopes P ⊆ [0, 1] n with P I = ∅ was established in Chvátal et al. [1989] . ...
For example, it is known that the Gomory-Chvátal rank of a polytope contained in the n-dimensional 0/1-cube is at most O(n 2 log n) Eisenbrand and Schulz [2003] , whereas the rank of all other methods ...
doi:10.1007/978-3-642-13036-6_34
fatcat:626wcllt6fbczbjdj5zv557zo4
Page 10498 of Mathematical Reviews Vol. , Issue 2004m
[page]
2004
Mathematical Reviews
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In this note, we show that the lower bound is also equal
to n. We relate this result to bounds on the disjunctive rank and
on the Lovasz-Schrijver rank of polytopes in the [0, 1]” cube. ...
Summary: “Eisenbrand and Schulz showed recently that the max-
imum Chvatal rank of a polytope in the [0, 1]” cube is bounded
above by O(n? logn) and bounded below by (1+¢)n for some € >
0. ...
0/1 Polytopes with Quadratic Chvatal Rank
[article]
2012
arXiv
pre-print
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We prove that there is a polytope contained in the 0/1 cube that has Chvatal rank Omega(n^2), closing the gap up to a logarithmic factor. ...
The number of iterations of this procedure that are needed until the integral hull of P is reached is called the Chvatal rank. ...
Acknowledgements The authors are very grateful to Michel X. Goemans for useful discussions. ...
arXiv:1204.4753v1
fatcat:36gamxed3zfevejlmsuzex6a24
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