0/1 Polytopes with Quadratic Chvatal Rank

TL;DR

This paper establishes a nearly tight quadratic lower bound on the Chvatal rank for certain 0/1 polytopes, advancing understanding of the complexity of cutting-plane procedures in integer programming.

Contribution

It proves a nearly tight quadratic lower bound on the Chvatal rank for 0/1 polytopes, closing a significant gap in the theoretical bounds.

Findings

Chvatal rank can be Omega(n^2) for some 0/1 polytopes

The proof links Chvatal rank to Diophantine approximation techniques

Provides new insights into the complexity of cutting-plane methods

Abstract

For a polytope P, the Chvatal closure P' is obtained by simultaneously strengthening all feasible inequalities cx <= b (with integral c) to cx <= floor(b). The number of iterations of this procedure that are needed until the integral hull of P is reached is called the Chvatal rank. If P is a subset of [0,1]^n, then it is known that O(n^2 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 Chvatal rank Omega(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 Chvatal rank to simultaneous Diophantine approximations w.r.t. the L1-norm of the normal vector defining P.