格的Voronoi胞的提升
Lifts for Voronoi cells of lattices
摘要：
在多面体组合学中出现的许多多面体是高维多面体的线性投影，具有显著较少的刻面。这样的提升可以产生多面体的压缩表示，其通常用于构造小尺寸线性规划。出于算法的影响，最近的向量问题，我们研究升降机的Voronoi细胞的格子。我们构造了一个明确的$d$维晶格，使得每个升降机各自的Voronoi细胞有$2^{Ω(d / \log d)}$方面。在积极的一面，我们表明，$d$维根晶格及其对偶晶格的Voronoi细胞有电梯与$O(d)$和$O(d \log d)$方面，分别。我们得到类似的结果spectrahedral电梯。
英文摘要：
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.
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