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 ${\mathcal L}$ outputs a set of points at least one of which is $\sqrt{\log n / n} \cdot \rho({\mathcal L})$-far from ${\mathcal L}$, where $\rho({\mathcal L})$ stands for the covering radius of ${\mathcal L}$ (i.e., the maximum possible distance of a point in space from ${\mathcal L}$). As an application, we show that the Covering Radius Problem with approximation factor $\sqrt{n /\log n}$ lies in the complexity class NP, improving a result of Guruswami, Micciancio and Regev by a factor of $\sqrt{\log n}$ (Computational Complexity, 2005).

Our results apply to any ℓ _p norm for 2 ≤ p ≤ ∞ with the same approximation factors (except a loss of $\sqrt{\log \log n}$ for p = ∞). In addition, we show that the output of our algorithm for RSP contains a point whose ℓ₂ distance from ${\mathcal L}$ is at least $(\log n / n)^{1/p} \cdot \rho^{(p)}({\mathcal L})$, where $\rho^{(p)}({\mathcal L})$ is the covering radius of ${\mathcal L}$ 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).