The gram-Schmidt walk: a cure for the banaszczyk blues

Nikhil Bansal; Shashwat Garg; Daniel Dadush; Shachar Lovett

doi:10.1145/3188745.3188850

The gram-Schmidt walk: a cure for the banaszczyk blues

Nikhil Bansal, Shashwat Garg, Daniel Dadush, Shachar Lovett

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

19 Citations (Scopus)

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.

Original language	English
Title of host publication	STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
Publisher	Association for Computing Machinery, Inc
Pages	1269-1282
Number of pages	14
ISBN (Electronic)	9781450355599
DOIs	https://doi.org/10.1145/3188745.3188850
Publication status	Published - 20 Jun 2018
Event	50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States Duration: 25 Jun 2018 → 29 Jun 2018

Conference

Conference	50th Annual ACM Symposium on Theory of Computing, STOC 2018
Country/Territory	United States
City	Los Angeles
Period	25/06/18 → 29/06/18

Keywords

Discrepancy
Random walks
Rounding techniques
rounding techniques
random walks

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10.1145/3188745.3188850

Cite this

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Bansal, N, Garg, S, Dadush, D & Lovett, S 2018, The gram-Schmidt walk: a cure for the banaszczyk blues. in STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. Association for Computing Machinery, Inc, pp. 1269-1282, 50th Annual ACM Symposium on Theory of Computing, STOC 2018, Los Angeles, United States, 25/06/18. https://doi.org/10.1145/3188745.3188850

The gram-Schmidt walk: a cure for the banaszczyk blues. / Bansal, Nikhil; Garg, Shashwat; Dadush, Daniel et al.
STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. Association for Computing Machinery, Inc, 2018. p. 1269-1282.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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The gram-Schmidt walk: a cure for the banaszczyk blues

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Keywords

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