A randomness-efficient sampler for matrix-valued functions and applications

Avi Wigderson*, David Xiao

*Corresponding author for this work

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

10 Scopus citations

Abstract

In this paper we give a randomness-efficient sampler for matrix-valued functions. Specifically, we show that a random walk on an expander approximates the recent Chernoff-like bound for matrix-valued functions of Ahlswede and Winter [1], in a manner which depends optimally on the spectral gap. The proof uses perturbation theory, and is a generalization of Gillman's and Lezaud's analyses of the Ajtai-Komlos-Szemeredi sampler for real-valued functions [11, 21, 2]. Derandomizing our sampler gives a few applications, yielding deterministic polynomial time algorithms for problems in which derandomizing independent sampling gives only quasi-polynomial time deterministic algorithms. The first (which was our original motivation) is to a polynomial-time derandomization of the Alon-Roichman theorem [4, 20, 22]: given a group of size n, find O(log n) elements which generate it as an expander. This implies a second application - efficiently constructing a randomness-optimal homomorphism tester, significantly improving the previous result of Shpilka and Wigderson [29]. A third application, which derandomizes a generalization of the set cover problem, is deferred to the full version of this paper.

Original languageEnglish
Title of host publicationProceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005
Pages397-406
Number of pages10
DOIs
StatePublished - 2005
Externally publishedYes
Event46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005 - Pittsburgh, PA, United States
Duration: 23 Oct 200525 Oct 2005

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2005
ISSN (Print)0272-5428

Conference

Conference46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005
Country/TerritoryUnited States
CityPittsburgh, PA
Period23/10/0525/10/05

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