A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
A lift-and-project cutting plane algorithm for mixed 0-1 programs
Mathematical Programming: Series A and B
Lower bounds for cutting planes proofs with small coefficients
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
On the Chvátal rank of polytopes in the 0/1 cube
Discrete Applied Mathematics
When Does the Positive Semidefiniteness Constraint Help in Lifting Procedures?
Mathematics of Operations Research
An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
On the Matrix-Cut Rank of Polyhedra
Mathematics of Operations Research
Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube
Combinatorica
Exponential Lower Bounds on the Lengths of Some Classes of Branch-and-Cut Proofs
Mathematics of Operations Research
Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Rank complexity gap for Lovász-Schrijver and Sherali-Adams proof systems
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Valid inequalities for mixed integer linear programs
Mathematical Programming: Series A and B
Integrality gaps of 2 - o(1) for Vertex Cover SDPs in the Lovész-Schrijver Hierarchy
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Linear Level Lasserre Lower Bounds for Certain k-CSPs
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Integrality gaps for Sherali-Adams relaxations
Proceedings of the forty-first annual ACM symposium on Theory of computing
Sherali-adams relaxations of the matching polytope
Proceedings of the forty-first annual ACM symposium on Theory of computing
Elementary closures for integer programs
Operations Research Letters
Design and verify: a new scheme for generating cutting-planes
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Random half-integral polytopes
Operations Research Letters
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We introduce a natural abstraction of propositional proof systems that are based on cutting planes. This new class of proof systems includes well-known operators such as Gomory-Chvátal cuts, lift-and-project cuts, Sherali-Adams cuts (for a fixed hierarchy level d), and split cuts. The rank of such a proof system corresponds to the number of rounds needed to show the nonexistence of integral solutions. We exhibit a family of polytopes without integral points contained in the n-dimensional 0/1-cube that has rank Ω(n/logn) for any proof system in our class. In fact, we show that whenever a specific cutting-plane based proof system has (maximal) rank n on a particular family of instances, then any cutting-plane proof system in our class has rank Ω(n/logn) for this family. This shows that the rank complexity of worst-case instances is intrinsic to the problem, and does not depend on specific cutting-plane proof systems, except for log factors. We also construct a new cutting-plane proof system that has worst-case rank O(n/logn) for any polytope without integral points, implying that the universal lower bound is essentially tight.