2012 IEEE 27th Conference on Computational Complexity (2011)

San Jose, California USA

June 8, 2011 to June 11, 2011

ISSN: 1093-0159

ISBN: 978-0-7695-4411-3

pp: 262-272

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2011.20

ABSTRACT

We construct a small set of explicit linear transformationsmapping R^n to R^t, where t=O(log (\gamma^{-1}) \eps^{-2} ), suchthat the L_2 norm of any vector in R^n is distorted by atmost 1\pm \eps in at least a fraction of 1-\gamma of thetransformations in the set. Albeit the tradeoff between thesize of the set and the success probability is sub-optimal comparedwith probabilistic arguments, we nevertheless are able to applyour construction to a number of problems. In particular, we use itto construct an \eps-sample (or pseudo-random generator) forlinear threshold functions on S^{n-1}, for \eps = o(1). Wealso use it to construct an \eps-sample for spherical digons inS^{n-1}, for \eps=o(1). This construction leads to anefficient oblivious derandomization of the Goemans-Williamson MAXCUT algorithm and similar approximation algorithms (i.e., weconstruct a small set of hyperplanes, such that for any instancewe can choose one of them to generate a good solution). Our technique for constructing \eps-sample for linear thresholdfunctions on the sphere is considerably different than previoustechniques that rely on k-wise independent sample spaces.

INDEX TERMS

Sample Space, Pseudo Random Generator, PRG, Johnoson-Lindenstrauss, Dimension Reduction, Derandomization, Max-Cut, Linear Threshold Function, Halfspace, Digon

CITATION

Amir Shpilka,
Zohar S. Karnin,
Yuval Rabani,
"Explicit Dimension Reduction and Its Applications",

*2012 IEEE 27th Conference on Computational Complexity*, vol. 00, no. , pp. 262-272, 2011, doi:10.1109/CCC.2011.20