Covering convex bodies and the Closest Vector Problem
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We present algorithms for the (1+ϵ)-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/ϵ)^n. We improve this substantially in the following two cases. For ℓ_p-norms with p>2 (resp. p ∈ [1,2]) fixed, we present an algorithm with a running time of 2^O(n) (1/ϵ)^n/2 (resp. 2^O(n) (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 two around their centroids, each one is contained in the (1+ϵ)-scaled homothet of the norm ball? We provide upper bounds for this problem by exploiting the modulus of smoothness of the ℓ_p-balls. Applying a covering scheme, we can boost any 2-approximation algorithm for CVP to a (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. Furthermore, we consider polyhedral and zonotopal norms. For centrally symmetric polytopes (resp. zonotopes) with O(n) facets (resp. generated by O(n) line segments), we provide a deterministic O(_2(1/ϵ))^O(n) time algorithm. This generalizes the result of Eisenbrand et al. which applies to the ℓ_∞-norm. As it is the case for the covering procedure of Eisenbrand et al., our approach boosts any constant factor approximation algorithm to a (1+ϵ)-approximate algorithm. By assuming the existence of a poly(n)-space and 2^O(n) time algorithm for 2-approximate CVP, the space complexity of our algorithm can be reduced to a polynomial.
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