The subdivision of large simplicial cones in Normaliz

TL;DR

This paper introduces improved algorithms for subdividing large simplicial cones in Normaliz, enhancing the computation of lattice points and Hilbert bases through integer programming and approximation techniques.

Contribution

It presents novel subdivision methods for large simplicial cones, integrating integer programming and approximation strategies to optimize Normaliz's performance.

Findings

Subdivision points found via integer programming improve efficiency.

Approximate overcones enable faster subdivision computations.

Enhanced algorithms handle large-volume cones more effectively.

Abstract

Normaliz is an open-source software for the computation of lattice points in rational polyhedra, or, in a different language, the solutions of linear diophantine systems. The two main computational goals are (i) finding a system of generators of the set of lattice points and (ii) counting elements degree-wise in a generating function, the Hilbert Series. In the homogeneous case, in which the polyhedron is a cone, the set of generators is the Hilbert basis of the intersection of the cone and the lattice, an affine monoid. We will present some improvements to the Normaliz algorithm by subdividing simplicial cones with huge volumes. In the first approach the subdivision points are found by integer programming techniques. For this purpose we interface to the integer programming solver SCIP to our software. In the second approach we try to find good subdivision points in an approximating overcone that is faster to compute.