Explicit dimension reduction and its applications

Zohar S. Karnin, Yuval Rabani, Amir Shpilka

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We construct a small set of explicit linear transformations mapping ℝn to ℝt, where t = O (log(γ-1) ε-2), such that the L2 norm of any vector in Rn is distorted by at most 1 ± ε in at least a fraction of 1 - γ of the transformations in the set. Albeit the tradeoff between the size of the set and the success probability is sub-optimal compared with probabilistic arguments, we nevertheless are able to apply our construction to a number of problems. In particular, we use it to construct an ε-sample (or pseudo-random generator) for linear threshold functions on Sn-1, for ε = o(1). We also use it to construct an ε-sample for spherical digons in Sn-1, for ε = o(1). This construction leads to an efficient oblivious derandomization of the Goemans-Williamson MAX CUT algorithm and similar approximation algorithms (i.e., we construct a small set of hyperplanes, such that for any instance we can choose one of them to generate a good solution). Our technique for constructing ε-sample for linear threshold functions on the sphere is considerably different than previous techniques that rely on k-wise independent sample spaces.

Original languageEnglish
Title of host publicationProceedings - 26th Annual IEEE Conference on Computational Complexity, CCC 2011
Pages262-272
Number of pages11
DOIs
StatePublished - 2011
Externally publishedYes
Event26th Annual IEEE Conference on Computational Complexity, CCC 2011 - San Jose, CA, United States
Duration: 8 Jun 201110 Jun 2011

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159

Conference

Conference26th Annual IEEE Conference on Computational Complexity, CCC 2011
Country/TerritoryUnited States
CitySan Jose, CA
Period8/06/1110/06/11

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