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Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube [chapter]

Friedrich Eisenbrand, Andreas S. Schulz
1999 Lecture Notes in Computer Science  
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]

Yohann Benchetrit, Samuel Fiorini, Tony Huynh, Stefan Weltge
2017 arXiv   pre-print
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

Sebastian Pokutta, Gautier Stauffer
2011 Operations Research Letters  
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

Sebastian Pokutta, Andreas S. Schulz
2011 Operations Research Letters  
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

Alexander Bockmayr, Friedrich Eisenbrand, Mark Hartmann, Andreas S. Schulz
1999 Discrete Applied Mathematics  
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

Yohann Benchetrit, Samuel Fiorini, Tony Huynh, Stefan Weltge
2018 Mathematics of Operations Research  
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  
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  
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  
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

Gábor Braun, Sebastian Pokutta
2011 Operations Research Letters  
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]

Thomas Rothvoß, Laura Sanitá
2013 Lecture Notes in Computer Science  
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

Thomas Rothvoß, Laura Sanità
2017 Operations Research  
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]

Sebastian Pokutta, Andreas S. Schulz
2010 Lecture Notes in Computer Science  
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  
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]

Thomas Rothvoss, Laura Sanita
2012 arXiv   pre-print
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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