The discrepancy of unsatisfiable matrices and a lower bound for the Komlós conjecture constant
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We construct simple, explicit matrices with columns having unit ℓ^2 norm and discrepancy approaching 1 + √(2)≈ 2.414. This number gives a lower bound, the strongest known as far as we are aware, on the constant appearing in the Komlós 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.
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