An important result in discrepancy due to Banaszczyk states that for any set of n vectors in Rm of 2 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.

Discrepancy, Random walks, Rounding techniques
Annual ACM SIGACT Symposium on Theory of Computing
Networks and Optimization

Bansal, N, Dadush, D.N, Garg, S, & Lovett, S. (2018). The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018 (pp. 587–597). doi:10.1145/3188745.3188850