### Abstract

An important result in discrepancy due to Banaszczyk States that for any set of n vectors in R^{m} of ℓ_{2} 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 |
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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 | |

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 |
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Country | United States |

City | Los Angeles |

Period | 25/06/18 → 29/06/18 |

### Keywords

- Discrepancy
- Random walks
- Rounding techniques
- rounding techniques
- random walks

## Cite this

*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