0/1 Polytopes with Quadratic Chvátal Rank

For a polytope P , the Chvátal closure P ′ ⊆ P is obtained by simultaneously strengthening all feasible inequalities cx ≤ β (with integral c) to cx ≤ ⌊β⌋. 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], then it is known that O(n log n) iterations always suffice (Eisenbrand and Schulz (1999)) and at least (1 + 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(n), 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 .