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2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Enumerative Lattice Algorithms in any Norm Via Mellipsoid Coverings
Palm Springs, California USA
October 22October 25
ISBN: 9780769545714
ASCII Text  x  
Daniel Dadush, Chris Peikert, Santosh Vempala, "Enumerative Lattice Algorithms in any Norm Via Mellipsoid Coverings," 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pp. 580589, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, 2011.  
BibTex  x  
@article{ 10.1109/FOCS.2011.31, author = {Daniel Dadush and Chris Peikert and Santosh Vempala}, title = {Enumerative Lattice Algorithms in any Norm Via Mellipsoid Coverings}, journal ={2013 IEEE 54th Annual Symposium on Foundations of Computer Science}, volume = {0}, year = {2011}, issn = {02725428}, pages = {580589}, doi = {http://doi.ieeecomputersociety.org/10.1109/FOCS.2011.31}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  CONF JO  2013 IEEE 54th Annual Symposium on Foundations of Computer Science TI  Enumerative Lattice Algorithms in any Norm Via Mellipsoid Coverings SN  02725428 SP580 EP589 A1  Daniel Dadush, A1  Chris Peikert, A1  Santosh Vempala, PY  2011 KW  Shortest/Closest Vector Problem KW  Integer Programming KW  Lattice Point Enumeration KW  Mellipsoid VL  0 JA  2013 IEEE 54th Annual Symposium on Foundations of Computer Science ER   
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2011.31
We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming (IP). Our enumeration technique relies on a classical concept from asymptotic convex geometry known as the Mellipsoid, and uses as a crucial subroutine the recent algorithm of Micciancio and Voulgaris (STOC 2010)for lattice problems in the l2 norm. As a main technical contribution, which may be of independent interest, we build on the techniques of Klartag (Geometric and Functional Analysis, 2006) to give an expected 2^O(n)time algorithm for computing an Mellipsoid for any ndimensional convex body. As applications, we give deterministic 2^O(n)time and space algorithms for solving exact SVP, and exact CVP when the target point is sufficiently close to the lattice, on ndimensional lattices in any (semi)norm given an Mellipsoid of the unit ball. In many norms of interest, including all lp norms, an Mellipsoid is computable in deterministic poly(n) time, in which case these algorithms are fully deterministic. Here our approach may be seen as a derandomization of the ââ‚¬Å“AKS sieveââ‚¬Âfor exact SVP and CVP (Ajtai, Kumar, and Siva Kumar, STOC2001 and CCC 2002). As a further application of our SVP algorithm, we derive an expected O(f*(n))^ntime algorithm for Integer Programming, where f*(n) denotes the optimal bound in the socalled ââ‚¬Å“flatnesstheorem, ââ‚¬Â which satisfies f*(n) = O(n^(4/3) polylog(n))and is conjectured to be f*(n) = O(n). Our runtime improves upon the previous best of O(n^2)^n by Hildebrand and Koppe(2010).
Index Terms:
Shortest/Closest Vector Problem, Integer Programming, Lattice Point Enumeration, Mellipsoid
Citation:
Daniel Dadush, Chris Peikert, Santosh Vempala, "Enumerative Lattice Algorithms in any Norm Via Mellipsoid Coverings," focs, pp.580589, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, 2011
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