Theory of linear and integer programming
Theory of linear and integer programming
Clique tree inequalities and the symmetric travelling salesman problem
Mathematics of Operations Research
On the complexity of cutting-plane proofs
Discrete Applied Mathematics
Facets of the three-index assignment polytope
Discrete Applied Mathematics
The Hirsch conjecture is true for (0,1)-polytopes
Mathematical Programming: Series A and B
Optimization
Small travelling salesman polytopes
Mathematics of Operations Research
Optimizing over the subtour polytope of the travelling salesman problem
Mathematical Programming: Series A and B
Anti-Hadamard matrices, coin weighing, threshold gates, and indecomposable hypergraphs
Journal of Combinatorial Theory Series A
Combinatorial optimization
Upper bounds on the maximal of facets of 0/1-polytopes
European Journal of Combinatorics - Special issue on combinatorics of polytopes
On the Chvátal Rank of Certain Inequalities
Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization
Operations Research Letters
On the Rank of Mixed 0, 1 Polyhedra
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
An Exponential Lower Bound on the Length of Some Classes of Branch-and-Cut Proofs
Proceedings of the 9th International IPCO Conference on Integer Programming and Combinatorial Optimization
Complexity of Semi-algebraic Proofs
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
Improving integrality gaps via Chvátal-Gomory rounding
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Elementary closures for integer programs
Operations Research Letters
0/1 polytopes with quadratic chvátal rank
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
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Gomory's and Chvátal's cutting-plane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvátal rank of the polyhedron. It is well-known that the Chvátal rank can be arbitrarily large, even if the polyhedron is bounded, if it is of dimension 2, and if its integer hull is a 0/1-polytope. We prove that the Chvátal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, the rank of any polytope contained in the n-dimensional 0/1-cube is at most 3n2 lg n. This improves upon a recent result of Bockmayr et al. [6] who obtained an upper bound of O(n3 lg n). Moreover, we refine this result by showing that the rank of any polytope in the 0/1-cube that is defined by inequalities with small coefficients is O(n). The latter observation explains why for most cutting planes derived in polyhedral studies of several popular combinatorial optimization problems only linear growth has been observed (see, e.g., [13]); the coefficients of the corresponding inequalities are usually small. Similar results were only known for monotone polyhedra before. Finally, we provide a family of polytopes contained in the 0/1-cube the Chvátal rank of which is at least (1+Ɛ)n for some Ɛ 0; the best known lower bound was n.