On the convergence of the affine hull of the Chv\'atal-Gomory closures

TL;DR

This paper studies the number of Chvatal-Gomory closure iterations needed to reach the affine hull of an integral polyhedron, showing bounds depend on the presence of interior integer points and the polyhedron's dimension.

Contribution

It establishes bounds on the iteration count for Chvatal-Gomory closures based on geometric properties of the polyhedron, highlighting cases with and without interior integer points.

Findings

Finite bounds exist if P contains an interior integer point.

No finite bound exists if P is not full-dimensional and lacks interior integer points.

Results depend solely on the dimension n of the space.

Abstract

Given an integral polyhedron P and a rational polyhedron Q living in the same n-dimensional space and containing the same integer points as P, we investigate how many iterations of the Chv\'atal-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.