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) (http://dx.doi.org/10.1137/1.9781611973105.78).
Additional Metadata
Keywords Approximation algorithms, Closest vector problem, High dimensional geometry, Lattice algorithms
ACM Nonnumerical Algorithms and Problems (acm F.2.2), Modes of Computation (acm F.1.2), Optimization (acm G.1.6)
MSC Approximation algorithms (msc 68W25), Analysis of algorithms and problem complexity (msc 68Q25), Randomized algorithms (msc 68W20)
THEME Logistics (theme 3)
Persistent URL dx.doi.org/10.4086/toc.2016.v012a002
Journal Theory of Computing
Citation
Dadush, D.N, & Kun, G. (2016). Lattice sparsification and the Approximate Closest Vector Problem. Theory of Computing, 12. doi:10.4086/toc.2016.v012a002

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