The discrepancy of unsatisfiable matrices and a lower bound for the Koml\'{o}s conjecture constant

Abstract

We construct simple, explicit matrices with columns having unit $\ell^2$ norm and discrepancy approaching $1 + \sqrt{2} \approx 2.414$. This number gives a lower bound, the strongest known as far as we are aware, on the constant appearing in the Koml\'{o}s conjecture. The "unsatisfiable matrices" giving this bound are built by scaling the entries of clause-variable matrices of certain unsatisfiable Boolean formulas. We show that, for a given formula, such a scaling maximizing a lower bound on the discrepancy may be computed with a convex second-order cone program. Using a dual certificate for this program, we show that our lower bound is optimal among those using unsatisfiable matrices built from formulas admitting read-once resolution proofs of unsatisfiability. We also conjecture that a generalization of this certificate shows that our bound is optimal among all bounds using unsatisfiable matrices.