The remote set problem on lattices

Abstract

We initiate studying the Remote Set Problem (RSP) on lattices, which given a lattice asks to find a set of points containing a point which is far from the lattice. We show a polynomial-time deterministic algorithm that on rank n lattice L outputs a set of points at least one of which is √log n/n·ρ(L)-far from L, where ρ(L) stands for the covering radius of L (i.e., the maximum possible distance of a point in space from L). As an application, we show that the Covering Radius Problem with approximation factor √log n lies in the complexity class NP, improving a result of Guruswami, Micciancio and Regev by a factor of √log n (Computational Complexity, 2005). Our results apply to any ℓ p norm for 2 ≤ p ≤ ∞ with the same approximation factors (except a loss of √ log log n for p = ∞). In addition, we show that the output of our algorithm for RSP contains a point whose ℓ 2 distance from L is at least (log n/n) 1/p·ρ (p)(L), where ρ (p)(L) is the covering radius of L measured with respect to the ℓ p norm. The proof technique involves a theorem on balancing vectors due to Banaszczyk (Random Struct. Alg., 1998) and the 'six standard deviations' theorem of Spencer (Trans. AMS, 1985). © 2012 Springer-Verlag.