Covering Convex Bodies and the Closest Vector Problem

Abstract

AbstractWe present algorithms for the $$(1+\epsilon )$$ ( 1 + ϵ ) -approximate version of the closest vector problem for certain norms. The currently fastest algorithm (Dadush and Kun 2016) for general norms in dimension n has running time of $$2^{O(n)}(1/\epsilon )^n$$ 2 O ( n ) ( 1 / ϵ ) n . We improve this substantially in the following two cases. First, for $$\ell _p$$ ℓ p -norms with $$p>2$$ p > 2 (resp. $$p \in [1,2]$$ p ∈ [ 1 , 2 ] ) fixed, we present an algorithm with a running time of $$2^{O(n)}(1+1/\epsilon )^{n/2}$$ 2 O ( n ) ( 1 + 1 / ϵ ) n / 2 (resp. $$2^{O(n)} (1+1/\epsilon )^{n/p}$$ 2 O ( n ) ( 1 + 1 / ϵ ) 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 factor 2 around their centroids, each one is contained in the $$(1+\epsilon )$$ ( 1 + ϵ ) -scaled homothet of the norm ball? We provide upper bounds for this $$(2,\epsilon )$$ ( 2 , ϵ ) -covering number by exploiting the modulus of smoothness of the $$\ell _p$$ ℓ p -balls. Applying a covering scheme, we can boost any 2-approximation algorithm for CVP to a $$(1+\epsilon )$$ ( 1 + ϵ ) -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. Second, we consider polyhedral and zonotopal norms. For centrally symmetric polytopes (resp. zonotopes) in $${\mathbb R}^n$$ R n with O(n) facets (resp. generated by O(n) line segments), we provide a deterministic $$O(\log _2(2+1/\epsilon ))^{O(n)}$$ O ( log 2 ( 2 + 1 / ϵ ) ) O ( n ) time algorithm. This generalizes the result of Eisenbrand et al. which applies to the $$\ell _\infty $$ ℓ ∞ -norm. Finally, we establish a connection between the modulus of smoothness and lattice sparsification. As a consequence, using the enumeration and sparsification tools developped by Dadush, Kun, Peikert, and Vempala, we present a simple alternative to the boosting procedure with the same time and space requirement for $$\ell _p$$ ℓ p norms. This connection might be of independent interest.