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 $cx \leq \lfloor\beta\rfloor$. 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(n ² 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(n ²), 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.