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 nonconstructive bound for the problem due to Banaszczyk. The previous algorithms only achieved an O(t1/2 log n) bound. The result also extends to the more general Komlós setting and gives an algorithmic O(log1/2n) bound.
Discrepancy, Random walk, Semidefinite programming
SIAM Journal on Computing
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Bansal, N, Dadush, D.N, & Garg, S. (2019). An algorithm for Komlós conjecture matching Banaszczyk's bound. SIAM Journal on Computing, 48(2), 534–553. doi:10.1137/17M1126795