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.
Many polytopes arising in polyhedral combinatorics are linear projections of higherdimensional 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.
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