On the Convergence of the Affine Hull of the Chvátal--Gomory Closures

Abstract

Given an integral polyhedron $P\subseteq\mathbb{R}^n$ and a rational polyhedron $Q\subseteq\mathbb{R}^n$ containing the same integer points as $P$, we investigate how many iterations of the Chvátal--Gomory closure operator have to be performed on $Q$ to obtain a polyhedron contained in the affine hull of $P$. We show that if $P$ contains an integer point in its relative interior, then such a number of iterations can be bounded by a function depending only on $n$. On the other hand, we prove that if $P$ is not full-dimensional and does not contain any integer point in its relative interior, then no finite bound on the number of iterations exists.