An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound

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(t
^1/2
log n) bound. Our result also extends to the more general Komlós setting and gives an algorithmic O(log
^1/2
n) bound.