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
On the Chvátal rank of polytopes in the 0/1 cube
Discrete Applied Mathematics
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube
Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization
An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0--1 Programming
Mathematics of Operations Research
Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube
Combinatorica
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
The Chvátal-Gomory Closure of a Strictly Convex Body
Mathematics of Operations Research
On the Chvátal-Gomory closure of a compact convex set
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Lower bounds for the Chvátal-Gomory rank in the 0/1 cube
Operations Research Letters
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
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For a polytope P, the Chvátal closureP′⊆P is obtained by simultaneously strengthening all feasible inequalities cx≤β (with integral c) to $cx \leq \lfloor\beta\rfloor$. The number of iterations of this procedure that are needed until the integral hull of P is reached is called the Chvátal rank. If P⊆[0,1]n, then it is known that O(n2 logn) iterations always suffice (Eisenbrand and Schulz (1999)) and at least $(1+\frac{1}{e}-o(1))n$ iterations are sometimes needed (Pokutta and Stauffer (2011)), leaving a huge gap between lower and upper bounds. We prove that there is a polytope contained in the 0/1 cube that has Chvátal rank Ω(n2), closing the gap up to a logarithmic factor. In fact, even a superlinear lower bound was mentioned as an open problem by several authors. Our choice of P is the convex hull of a semi-random Knapsack polytope and a single fractional vertex. The main technical ingredient is linking the Chvátal rank to simultaneous Diophantine approximations w.r.t. the ∥·∥1-norm of the normal vector defining P.