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 $O(\sqrt{n})$. 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 $n \times n$ matrices that are block diagonal with block size q admit a coloring of discrepancy $O(\sqrt{n} \cdot \sqrt{\log (q)})$. Bansal, Dadush and Garg recently gave a randomized algorithm to find a vector x with entries in $\lbrace {-1,1\rbrace }$ with $\Vert Ax\Vert _{\infty } \le O(\sqrt{\log n})$ in polynomial time, where A is any matrix whose columns have length at most 1. We show that our method can be used to deterministically obtain such a vector.

T. Rothvoss—Supported by NSF grant 1420180 with title “Limitations of convex relaxations in combinatorial optimization”, an Alfred P. Sloan Research Fellowship and a David & Lucile Packard Foundation Fellowship.