0/1 Polytopes with Quadratic Chvátal Rank

Abstract

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]n, then it is known that O(n2 log n) iterations always suffice and at least (1 + 1/e − o(1))n iterations are sometimes needed, 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.