Covering Cubes and the Closest Vector Problem

TL;DR

This paper introduces a faster randomized (1+epsilon)-approximation algorithm for the closest vector problem in the infinity norm, utilizing geometric covering techniques with ellipsoids and parallelepipeds to improve computational efficiency.

Contribution

It presents the fastest known randomized approximation algorithm for CVP in infinity norm, based on novel geometric covering bounds and a method to enhance existing 2-approximation algorithms.

Findings

Achieves a running time of 2^O(n) (log 1/epsilon)^O(n), improving previous bounds.

Provides an almost optimal bound for covering cubes with axis-parallel ellipsoids.

Develops a scheme to convert 2-approximation algorithms into (1+epsilon)-approximation algorithms.

Abstract

We provide the currently fastest randomized (1+epsilon)-approximation algorithm for the closest vector problem in the infinity norm. The running time of our method depends on the dimension n and the approximation guarantee epsilon by 2^O(n) (log 1/epsilon)^O(n)$ which improves upon the (2+1/epsilon)^O(n) running time of the previously best algorithm by Bl\"omer and Naewe. Our algorithm is based on a solution of the following geometric covering problem that is of interest of its own: Given epsilon in (0,1), how many ellipsoids are necessary to cover the cube [-1+epsilon, 1-epsilon]^n such that all ellipsoids are contained in the standard unit cube [-1,1]^n? We provide an almost optimal bound for the case where the ellipsoids are restricted to be axis-parallel. We then apply our covering scheme to a variation of this covering problem where one wants to cover [-1+epsilon,1-epsilon]^n with parallelepipeds that, if scaled by two, are still contained in the unit cube. Thereby, we obtain a method to boost any 2-approximation algorithm for closest-vector in the infinity norm to a (1+epsilon)-approximation algorithm that has the desired running time.