An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound

Abstract

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) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> ), matching the best known non-constructive bound for the problem due to Banaszczyk. The previous algorithms only achieved an O(t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> log n) bound. Our result also extends to the more general Komlós setting and gives an algorithmic O(log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> n) bound.