A polynomial time algorithm for solving the closest vector problem in zonotopal lattices

Abstract

In this note we give a polynomial time algorithm for solving the closest vector problem in the class of zonotopal lattices. The Voronoi cell of a zonotopal lattice is a zonotope, i.e. a projection of a regular cube. Examples of zonotopal lattices include lattices of Voronoi's first kind and tensor products of root lattices of type A. The combinatorial structure of zonotopal lattices can be described by regular matroids/totally unimodular matrices. We observe that a linear algebra version of the minimum mean cycle canceling method can be applied for efficiently solving the closest vector problem in a zonotopal lattice if the lattice is given as the integral kernel of a totally unimodular matrix.