Hardness of Approximating the Closest Vector Problem with Pre-Processing

Authors:
Mikhail Alekhnovich;Subhash A. Khot;Guy Kindler;Nisheeth K. Vishnoi
Affiliations:
Mikhail Alekhnovich;Subhash A. Khot;-;Nisheeth K. Vishnoi
Venue:
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Year:
2005

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Abstract

We show that, unless NP\subseteqDTIME(2^{poly\log (n)}), the closest vector problem with pre-processing, for \ell \rho norm for any p \ge 1, is hard to approximate within a factor of (\log n)^{1/p - \ell } for any \varepsilon 0. This improves the previous best factor of 3^{1/p} - \varepsilon due to Regev [19]. Our results also imply that under the same complexity assumption, the nearest codeword problem with pre-processing is hard to approximate within a factor of (\log n)^{1 - \varepsilon }for any \varepsilon 0.