Vector Balancing in Polynomial Time

Abstract

We present a spectral signing algorithm solving the Koml\'os problem with a constant discrepancy in polynomial time. Given a matrix $A\in\mathbb{R}^{m\times n}$ whose columns have Euclidean norm at most $1$, the algorithm finds a vector $\varepsilon\in\{-1,1\}^n$ satisfying $\|A\varepsilon\|_\infty\le C$, where $C$ is an absolute constant. By minimizing a cubic spectral potential, our spectral signing algorithm updates the fractional coloring toward Boolean signs with time complexity $O((mn^9+n^{10})\log(2+m+n))$.