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.
翻译:我们构建了简单、清晰的矩阵, 列内有单位 $\ $2, 标准值和差异值, 接近 1 +\ sqrt{2}\ approx 2. 414$。 这个数字在 Koml\\ { o} 的猜想中显示的恒定值上, 给出这一约束值的“ 无法满足的矩阵” 是用某些不满意的布尔林公式的可条款可变矩阵条目的缩放来构建的。 我们显示, 对于给定公式来说, 将差异的下限最大化, 可以用一个 comvex 二阶锥程序来计算。 我们使用此程序的双轨证书, 显示我们的下限值在使用接受不满足性公式的不可满足性矩阵中是最佳的。 我们还推测, 该证书的概括化显示, 我们的约束值是使用不满足性矩阵在所有约束值中的最佳值 。