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Covering Cubes and the Closest Vector Problem

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 Blömer and Naewe. Our algorithm is based on a solution of the following geometric covering problem that is of interest of its own: Given epsilon>0, how many ellipsoids are necessary to cover the scaled unit cube [-1+epsilon, 1-epsilon]^n such 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 the scaled cube with boxes 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.

Published in:

Computational Geometry (Scg 11), 417-423

Presented at:

27th Annual Symposium on Computational Geometry (SoCG 2011), Paris, France, June 13-15, 2011

Publisher:

Acm Order Department, P O Box 64145, Baltimore, Md 21264 Usa

ISBN:

978-1-4503-0682-9

Keywords:

Laboratories:

Record created 2011-02-15, last modified 2019-03-16