Covering convex bodies and the Closest Vector Problem

We present algorithms for the $(1+\epsilon)$-approximate version of the closest vector problem for certain norms. The currently fastest algorithm (Dadush and Kun 2016) for general norms has running time of $2^{O(n)} (1/\epsilon)^n$. We improve this substantially in the following two cases. For $\ell_p$-norms with $p>2$ (resp. $p \in [1,2]$) fixed, we present an algorithm with a running time of $2^{O(n)} (1/\epsilon)^{n/2}$ (resp. $2^{O(n)} (1/\epsilon)^{n/p}$). This result is based on a geometric covering problem, that was introduced in the context of CVP by Eisenbrand et al.: How many convex bodies are needed to cover the ball of the norm such that, if scaled by two around their centroids, each one is contained in the $(1+\epsilon)$-scaled homothet of the norm ball? We provide upper bounds for this problem by exploiting the \emph{modulus of smoothness} of the $\ell_p$-balls. Applying a covering scheme, we can boost any constant approximation algorithm for CVP to a $(1+\epsilon)$-approximation algorithm with the improved run time, either using a straightforward sampling routine or using the deterministic algorithm of Dadush for the construction of an epsilon net. The space requirement only depends on the constant approximation CVP solver used. Furthermore, we generalise the result of Eisenbrand et al. for the $\ell_\infty$-norm. For centrally symmetric polytopes (resp. zonotopes) with $O(n)$ facets (resp. generated by $O(n)$ line segments), we provide a deterministic $O(\log_2(1/\epsilon))^{O(n)}$ time algorithm. Finally, we establish a connection between the \emph{modulus of smoothness} and \emph{lattice sparsification}. Using the enumeration and sparsification tools developped by Dadush, Kun, Peikert and Vempala, this leads to a simple alternative to the boosting procedure for CVP under $\ell_p$-norms. This connection might be of independent interest.