Deterministic discrepancy minimization via the multiplicative weight update method

Abstract

A well-known theorem of Spencer shows that any set system with n sets over n elements admits a coloring of discrepancy O(√n). While the original proof was non-constructive, recent progress brought polynomial time algorithms by Bansal, Lovett and Meka, and Rothvoss. All those algorithms are randomized, even though Bansal’s algorithm admitted a complicated derandomization. We propose an elegant deterministic polynomial time algorithm that is inspired by Lovett-Meka as well as the Multiplicative Weight Update method. The algorithm iteratively updates a fractional coloring while controlling the exponential weights that are assigned to the set constraints. A conjecture by Meka suggests that Spencer’s bound can be generalized to symmetric matrices. We prove that n × n matrices that are block diagonal with block size q admit a coloring of discrepancy O(√n · √log(q)). Bansal, Dadush and Garg recently gave a randomized algorithm to find a vector x with entries in {−1, 1} with ‖Ax‖∞ ≤ O(√log n) in polynomial time, where A is any matrix whose columns have length at most 1. We show that our method can be used to deter-ministically obtain such a vector.