Komlós conjecture (constant ℓ∞ discrepancy for unit-norm columns)
Establish the Komlós conjecture: Determine a universal constant C such that for every m×n real matrix M whose columns have ℓ2-norm at most 1, there exist signs x in {−1,1}^n with ||Mx||_∞ ≤ C, independent of m and n.
References
The central open problems in this area are the Beck-Fiala conjecture and more generally the Komlos conjecture, where given a set of $n$ vectors in the unit ball in $m$, the goal is to assign signs (also called ``colors") to the vectors so as to bound the $\ell_{\infty}$ norm of the final signed sum, by a constant independent of $m$ and $n$.
Find a deterministic algorithm for the Komlós problem with $O(mn+n{\omega+2-c})$ operations for a constant $c>0$, or with $o(n2)$ updates. The example above suggests what is needed: a potential that controls the averages of $W$ over every set of active coordinates and is smooth at a scale coarser than $\eta$.
Find a polynomial-time algorithm with discrepancy at most $36$, the bound of the elementary proof, or even $3\sqrt{2\pi}$; an efficient Banaszczyk-type induction is one route.