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 proceedingConference contributionAcademicpeer-review

5 Citations (Scopus)

Abstract

An important result in discrepancy due to Banaszczyk States that for any set of n vectors in Rm of ℓ2 norm at most 1 and any convex body K in Rm 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 languageEnglish
Title of host publicationSTOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
PublisherAssociation for Computing Machinery, Inc
Pages1269-1282
Number of pages14
ISBN (Electronic)9781450355599
DOIs
Publication statusPublished - 20 Jun 2018
Event50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States
Duration: 25 Jun 201829 Jun 2018

Conference

Conference50th Annual ACM Symposium on Theory of Computing, STOC 2018
CountryUnited States
CityLos Angeles
Period25/06/1829/06/18

Keywords

  • Discrepancy
  • Random walks
  • Rounding techniques
  • rounding techniques
  • random walks

Cite this

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 (pp. 1269-1282). Association for Computing Machinery, Inc. https://doi.org/10.1145/3188745.3188850