A Randomness-Efficient Sampler for Matrix-valued Functions and Applications

  • Authors:

  • Avi Wigderson;David Xiao

  • Affiliations:

  • Institute for Advanced Study, Princeton Univeristy;Princeton Univeristy

  • Venue:
  • FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
  • Year:
  • 2005

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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驴s and Lezaud驴s analyses of the Ajtai-Komlos-Szemeredi sampler for realvalued 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 polynomialtime derandomization of the Alon-Roichmantheorem [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.