On the Computation of Hilbert Bases and Extreme Rays of Cones

TL;DR

This paper introduces a new project-and-lift method for efficiently computing Hilbert bases and extreme rays of cones, with applications in combinatorics and computational algebra.

Contribution

It presents a novel approach that improves the computation of minimal generators and extreme rays of cones associated with lattices and integer matrices.

Findings

Effective computation of Hilbert bases demonstrated

Method applied successfully to combinatorial problems

Computational experiments show improved efficiency

Abstract

In this paper we present a novel project-and-lift approach to compute the set of minimal generators of the semigroup $(\Lambda\cap\R^n_+,+)$ for lattices $\Lambda\subseteq\Z^n$. This problem class includes the computation of Hilbert bases of cones $\{z:Az=0,z\in\R^n_+\}$ for integer matrices $A$. A similar approach can be used to compute only the extreme rays of such cones. Finally, some combinatorial applications and computational experience are presented.