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arXiv · 2106.04432

Lifts for Voronoi cells of lattices

Matthias Schymura·Ina Seidel·Stefan Weltge
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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 $2^{\Omega(d / \log d)}$ 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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Matthias Schymura, Ina Seidel, Stefan Weltge. 2021-06-08. Lifts for Voronoi cells of lattices. https://arxiv.org/abs/2106.04432

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