The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues

2017/08/03 by Bansal, Nikhil, Dadush, Daniel, Garg, Shashwat +1 · 3 citations
#Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences

paper · doi:10.48550/arxiv.1708.01079

Abstract

An important result in discrepancy due to Banaszczyk states that for any set of n vectors in ℝ^m of ℓ₂ norm at most 1 and any convex body K in ℝ^m of Gaussian measure at least half, there exists a ± 1 combination of these vectors which lies in 5K. This result implies the best known bounds for several problems in discrepancy. Banaszczyk's proof of this result is non-constructive and a major open problem has been to give an efficient algorithm to find such a ± 1 combination of the vectors. In this paper, we resolve this question and give an efficient randomized algorithm to find a ± 1 combination of the vectors which lies in cK for c>0 an absolute constant. This leads to new efficient algorithms for several problems in discrepancy theory.

Cited by

• Monotonicity of the Gaussian measure under Banaszczyk transforms
• Decoupling via Affine Spectral-Independence: Beck-Fiala and Komlós Bounds Beyond Banaszczyk
• An Improved Bound for the Beck-Fiala Conjecture

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