Lifts for Voronoi cells of lattices

06/08/2021

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by   Matthias Schymura, et al.

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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^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 O(d) and O(d log d) facets, respectively. We obtain similar results for spectrahedral lifts.