Iterated Chvatal-Gomory Cuts and the Geometry of Numbers

TL;DR

This paper investigates the strength of iterated Chvatal-Gomory cuts in integer programming, proposing an empirical method for selecting strong cuts and a polynomial-time algorithm for finding such cuts using geometry of numbers techniques.

Contribution

It introduces an empirical approach to identify strong iterates of CG-cuts and presents a polynomial-time algorithm for finding strong cuts based on geometry of numbers.

Findings

Empirical results show a specific iterate selection method outperforms others.

A polynomial-time algorithm is provided for finding strong CG-cut iterates.

The approach leverages the geometry of numbers to improve cut strength.

Abstract

Chvatal-Gomory cutting planes (CG-cuts for short) are a fundamental tool in Integer Programming. Given any single CG-cut, one can derive an entire family of CG-cuts, by `iterating' its multiplier vector modulo one. This leads naturally to two questions: first, which iterates correspond to the strongest cuts, and, second, can we find such strong cuts efficiently? We answer the first question empirically, by showing that one specific approach for selecting the iterate tends to perform much better than several others. The approach essentially consists in solving a nonlinear optimization problem over a special lattice associated with the CG-cut. We then provide a partial answer to the second question, by presenting a polynomial-time algorithm that yields an iterate that is strong in a certain well-defined sense. The algorithm is based on results from the algorithmic geometry of numbers.