Volume 15 (2019) Article 21 pp. 1-27
The Gram--Schmidt Walk: A Cure for the Banaszczyk Blues
Revised: December 13, 2019
Published: December 23, 2019
A classic result of Banaszczyk (Random Str. & Algor. 1997) states that given any $n$ vectors in $\mathbb{R}^m$ with $\ell_2$-norm at most $1$ and any convex body $K$ in $\mathbb{R}^m$ of Gaussian measure at least half, there exists a $\pm 1$ combination of these vectors that lies in $5K$. Banaszczyk's proof of this result was non-constructive and it was open how to find such a $\pm 1$ combination in polynomial time. In this paper, we give an efficient randomized algorithm to find a $\pm 1$ combination of the vectors which lies in $cK$ for some fixed constant $c> 0$. This leads to new efficient algorithms for several problems in discrepancy theory.