How to Find the Convex Hull of All Integer Points in a Polyhedron?

TL;DR

This paper introduces DDM Cuts, a novel cut-based algorithm leveraging Gomory cuts and dynamic double description for efficiently computing the convex hull of all integer points in a polyhedron, with promising computational results.

Contribution

The paper presents a new algorithm, DDM Cuts, that combines Gomory cuts with a dynamic double description method to improve convex hull computation of integer points.

Findings

DDM Cuts outperforms naive algorithms in computational experiments.

The algorithm effectively finds all vertices and facets of the integer convex hull.

Implementation details demonstrate practical applicability.

Abstract

We propose a cut-based algorithm for finding all vertices and all facets of the convex hull of all integer points of a polyhedron defined by a system of linear inequalities. Our algorithm DDM Cuts is based on the Gomory cuts and the dynamic version of the double description method. We describe the computer implementation of the algorithm and present the results of computational experiments comparing our algorithm with a naive one.