Lifts for Voronoi cells of lattices

TL;DR

This paper investigates the complexity of lifting Voronoi cells of lattices, revealing exponential lower bounds for some lattices and efficient lifts for root lattices, with implications for polyhedral representations.

Contribution

It provides explicit constructions showing exponential facet complexity for lifts of certain Voronoi cells and identifies classes of lattices with efficient lifts, advancing understanding of polyhedral representations.

Findings

Constructed a lattice with Voronoi cell lifts having exponential facets.

Root lattice Voronoi cells have lifts with linear or near-linear facets.

Results impact polyhedral approaches to lattice problems.

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.