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Volume 12 (2016) Article 2 pp. 1-34
Lattice Sparsification and the Approximate Closest Vector Problem
Received: July 11, 2013
Revised: January 11, 2016
Published: June 16, 2016
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Keywords: approximation algorithms, high-dimensional geometry, lattice algorithms, Closest Vector Problem
ACM Classification: F.2.2, F.1.2, G.1.6
AMS Classification: 68W25, 68Q25, 68W20

Abstract: [Plain Text Version]

$ \newcommand{\eps}{\varepsilon} \newcommand{\poly}{{\rm poly}} $

We give a deterministic algorithm for solving the $(1+\eps)$-approximate Closest Vector Problem (CVP) on any $n$-dimensional lattice and in any near-symmetric norm in $2^{O(n)}(1+1/\eps)^n$ time and $2^n\poly(n)$ space. Our algorithm builds on the lattice point enumeration techniques of Micciancio and Voulgaris (STOC 2010, SICOMP 2013) and Dadush, Peikert and Vempala (FOCS 2011), and gives an elegant, deterministic alternative to the “AKS Sieve”-based algorithms for $(1+\eps)$-CVP (Ajtai, Kumar, and Sivakumar; STOC 2001 and CCC 2002). Furthermore, assuming the existence of a $\poly(n)$-space and $2^{O(n)}$-time algorithm for exact CVP in the $\ell_2$ norm, the space complexity of our algorithm can be reduced to polynomial.

Our main technical contribution is a method for “sparsifying” any input lattice while approximately maintaining its metric structure. To this end, we employ the idea of random sublattice restrictions, which was first employed by Khot (FOCS 2003, J. Comp. Syst. Sci. 2006) for the purpose of proving hardness for the Shortest Vector Problem (SVP) under $\ell_p$ norms.

A preliminary version of this paper appeared in the Proc. 24th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA'13).