Reverse Chv\'atal-Gomory rank

TL;DR

This paper introduces the concept of reverse Chvátal-Gomory rank for integral polyhedra, characterizes those with infinite rank, and explores bounds when the rank is finite, advancing understanding of polyhedral integer programming.

Contribution

It defines the reverse Chvátal-Gomory rank, provides a geometric characterization for infinite cases, and investigates bounds for finite cases, offering new insights into polyhedral ranks.

Findings

Existence of integral polytopes with infinite reverse CG rank in dimension two

Geometric characterization of polyhedra with infinite reverse CG rank

Upper bounds on reverse CG rank when finite

Abstract

We introduce the reverse Chv\'atal-Gomory rank r*(P) of an integral polyhedron P, defined as the supremum of the Chv\'atal-Gomory ranks of all rational polyhedra whose integer hull is P. A well-known example in dimension two shows that there exist integral polytopes P with r*(P) equal to infinity. We provide a geometric characterization of polyhedra with this property in general dimension, and investigate upper bounds on r*(P) when this value is finite.