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Complexity and algorithms for computing Voronoi cells of lattices

by Mathieu Dutour Sikiric and Achill Schürmann and Frank Vallentin.

Series: 2008-19, Preprints

03D15 Complexity of computation (including implicit computational complexity)
11H56 Automorphism groups of lattices
11H06 Lattices and convex bodies
52B12 Special polytopes (linear programming, centrally symmetric, etc.)

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 \sharpp-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.

lattice, Voronoi cell, Delone cell, covering radius, quantizer constant, lattice isomorphism problem, zonotope