Complexity and algorithms for computing Voronoi cells of lattices

TL;DR

This paper investigates the computational complexity of Voronoi cell vertices in Euclidean lattices, proving #P-hardness, and introduces an efficient algorithm for low-dimensional, highly-symmetric cases, with practical applications to prominent lattices.

Contribution

It establishes the #P-hardness of computing Voronoi cell vertices and presents a specialized algorithm that outperforms existing tools for certain lattice classes.

Findings

Computing the number of Voronoi cell vertices is #P-hard.

The proposed algorithm is efficient for dimensions up to 12 and highly-symmetric lattices.

Applied algorithm to find vertices and quantizer constants of notable lattices.

Abstract

In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean lattice. Given a basis of a lattice, we prove that computing the number of vertices is a #P-hard problem. On the other hand we describe an algorithm for this problem which is especially suited for low dimensional (say dimensions at most 12) and for highly-symmetric lattices. We use our implementation, which drastically outperforms those of current computer algebra systems, to find the vertices of Voronoi cells and quantizer constants of some prominent lattices.