We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most t sets. We give an efficient algorithm that finds a coloring with discrepancy O((t log n)1/2), matching the best known non-constructive bound for the problem due to Banaszczyk. The previous algorithms only achieved an O(t1/2 log n) bound. Our result also extends to the more general Komlós setting and gives an algorithmic O(log1/2 n) bound.

Annual IEEE Symposium on Foundations of Computer Science
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Bansal, N., Dadush, D., & Garg, S. (2016). An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS (pp. 788–799). doi:10.1109/FOCS.2016.89