Deterministic Discrepancy Minimization via the Multiplicative Weight Update Method
Abstract
A wellknown 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 nonconstructive, 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 LovettMeka 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 $\Ax\_{\infty} \leq O(\sqrt{\log n})$ 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2016
 arXiv:
 arXiv:1611.08752
 Bibcode:
 2016arXiv161108752L
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Computational Geometry;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics
 EPrint:
 16 pages