Improved Algorithmic Bounds for Discrepancy of Sparse Set Systems

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 algorithm that finds a coloring with discrepancy O((t log n log s)) where s is the maximum cardinality of a set. This improves upon the previous constructive bound of O(t logn) based on algorithmic variants of the partial coloring method, and for small s (e.g. s = poly(t)) comes close to the non-constructive O((t log n)) bound due to Banaszczyk. Previously, no algorithmic results better than O(t logn) were known even for s = O(t). Our method is quite robust and we give several refinements and extensions. For example, the coloring we obtain satisfies the stronger size-sensitive property that each set S in the set system incurs an O((t log n log |S|)1/2) discrepancy. Another variant can be used to essentially match Banaszczyk’s bound for a wide class of instances even where s is arbitrarily large. Finally, these results also extend directly to the more general Komlós setting.