The remote set problem on lattices

Abstract

We initiate studying the Remote Set Problem ( $${\mathsf{RSP}}$$ 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 $${\mathcal{L}}$$ L outputs a set of points, at least one of which is $${\sqrt{\log n / n} \cdot \rho(\mathcal{L})}$$ log n / n · ? ( L ) -far from $${\mathcal{L}}$$ L , where $${\rho(\mathcal{L})}$$ ? ( L ) stands for the covering radius of $${\mathcal{L}}$$ L (i.e., the maximum possible distance of a point in space from $${\mathcal{L}}$$ L ). As an application, we show that the covering radius problem with approximation factor $${\sqrt{n / \log n}}$$ n / log n lies in the complexity class $${\mathsf{NP}}$$ NP , improving a result of Guruswami et al. (Comput Complex 14(2): 90---121, 2005) by a factor of $${\sqrt{\log n}}$$ log n . Our results apply to any $${\ell_p}$$ l p norm for $${2 \leq p \leq \infty}$$ 2 ≤ p ≤ ? with the same approximation factors (except a loss of $${\sqrt{\log \log n}}$$ log log n for $${p = \infty}$$ p = ? ). In addition, we show that the output of our algorithm for $${\mathsf{RSP}}$$ RSP contains a point whose $${\ell_2}$$ l 2 distance from $${\mathcal{L}}$$ L is at least $${(\log n/n)^{1/p} \cdot \rho^{(p)}(\mathcal{L})}$$ ( log n / n ) 1 / p · ? ( p ) ( L ) , where $${\rho^{(p)}(\mathcal{L})}$$ ? ( p ) ( L ) is the covering radius of $${\mathcal{L}}$$ L measured with respect to the $${\ell_p}$$ l p norm. The proof technique involves a theorem on balancing vectors due to Banaszczyk (Random Struct Algorithms 12(4):351---360, 1998) and the "six standard deviations" theorem of Spencer (Trans Am Math Soc 289(2):679---706, 1985).