Lifts for Voronoi Cells of Lattices

Abstract

Many polytopes arising in polyhedral combinatorics are linear projections of higher-dimensional polytopes with significantly fewer facets. Such lifts may yield compressed representations of polytopes, which are typically used to construct small-size linear programs. Motivated by algorithmic implications for the closest vector problem, we study lifts of Voronoi cells of lattices. We construct an explicit d-dimensional lattice such that every lift of the respective Voronoi cell has $$2^{\Omega (d/{\log d})}$$ 2 Ω ( d / log d ) facets. On the positive side, we show that Voronoi cells of d-dimensional root lattices and their dual lattices have lifts with $${{\mathcal {O}}}(d)$$ O ( d ) and $${{\mathcal {O}}}(d \log d)$$ O ( d log d ) facets, respectively. We obtain similar results for spectrahedral lifts.