An Improved Bound for the Beck-Fiala Conjecture

TL;DR

This paper presents an improved algorithmic bound for the Beck-Fiala conjecture, reducing the discrepancy from previous bounds and nearly matching the conjecture's predicted bound in a broad range of parameters.

Contribution

It provides an algorithmic proof achieving a discrepancy bound of O(√(k log log n)) for large k, significantly improving previous bounds.

Findings

Achieves discrepancy bound of O(√(k log log n)) for k ≥ log^5 n

Matches the Beck-Fiala conjecture up to a factor of O(√log log n)

Improves upon previous bounds of O(k) and O(√(k log n))

Abstract

In 1981, Beck and Fiala [Discrete Appl. Math, 1981] conjectured that given a set system $A \in \{0,1\}^{m \times n}$ with degree at most $k$ (i.e., each column of $A$ has at most $k$ non-zeros), its combinatorial discrepancy $\mathsf{disc}(A) := \min_{x \in \{\pm 1\}^n} \|Ax\|_\infty$ is at most $O(\sqrt{k})$. Previously, the best-known bounds for this conjecture were either $O(k)$, first established by Beck and Fiala [Discrete Appl. Math, 1981], or $O(\sqrt{k \log n})$, first proved by Banaszczyk [Random Struct. Algor., 1998]. We give an algorithmic proof of an improved bound of $O(\sqrt{k \log\log n})$ whenever $k \geq \log^5 n$, thus matching the Beck-Fiala conjecture up to $O(\sqrt{\log \log n})$ for almost the full regime of $k$.