发表机构
University of New South Wales（新南威尔士大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Komlós猜想，提出一种确定性快速谱符号化算法，在$O(mn+n^{\omega+2}\log^3 n)$时间内找到差异小于99的符号，改进了先前算法的运行时间与差异界。
AI 中文摘要
Komlós猜想（现已成为定理）断言：若矩阵$A\in\mathbb{R}^{m\times n}$的每一列的欧几里得范数至多为1，则存在符号$\varepsilon\in\{-1,1\}^n$使得$A\varepsilon$的每个坐标都被一个绝对常数所界定。Guo、Fang和Lu给出了第一个多项式时间算法来寻找这样的符号，这是一种确定性的谱符号化过程，其差异为8272，运行时间为$O((mn^9+n^{10})\log(m+n))$。我们给出一种确定性算法，使用$O(mn+n^{\omega+2}\log^3 n)$次算术运算找到满足$\\|A\varepsilon\\|_\infty<99$的符号，其中$\omega>2$是任意固定的可达矩阵乘法指数；根据当前$\omega$的界，这相当于$\widetilde O(mn+n^{4.372})$。我们的算法使用相同的框架：它对一个分数着色进行舍入，并通过能量校正势垒的Gram矩阵的最大特征值来监视所有行。步骤沿平坦方向进行，并按比例缩放，使得任何接近其阈值的势垒移动速度不超过常数，且当坐标冻结时正则化器增长；这些共同将更新次数限制为$O(n^2\log n)$。对跟踪的行和施加一个弱二次电荷，使得在任何时刻只需评估$O(n\log^2 n)$行，而运动时钟限制了任何其他行可能接近其势垒的时间。每次更新是一系列短矩阵乘积。其方向通过多项式软投影仪的条件期望读出，其小的约束残差在仿射行表示中修复，并且一个恒等式解释了每次表示的变化。对于有理数输入，该算法具有多项式位复杂度。
英文摘要
The Komlós conjecture, now a theorem, asserts that whenever the columns of a matrix $A\in\mathbb{R}^{m\times n}$ have Euclidean norm at most one, some signs $\varepsilon\in\{-1,1\}^n$ make every coordinate of $A\varepsilon$ bounded by an absolute constant. Guo, Fang, and Lu gave the first polynomial-time algorithm for finding such signs, a deterministic spectral signing procedure with discrepancy $8272$ and running time $O((mn^9+n^{10})\log(m+n))$. We give a deterministic algorithm that finds signs with $\|A\varepsilon\|_\infty<99$ using $O(mn+n^{ω+2}\log^3 n)$ arithmetic operations, where $ω>2$ is any fixed attainable matrix-multiplication exponent; with the current bounds on $ω$ this is $\widetilde O(mn+n^{4.372})$. Our algorithm uses the same framework: it rounds a single fractional coloring and watches all rows through the top eigenvalue of a Gram matrix of energy-corrected barriers. Steps follow flat directions, rescaled so that no barrier near its threshold moves faster than a constant, and the regularizer grows as coordinates freeze; together these bound the number of updates by $O(n^2\log n)$. A weak quadratic charge on the tracked row sums leaves $O(n\log^2 n)$ rows to evaluate at any time, and a motion clock bounds when any other row could approach its barrier. Each update is a short sequence of matrix products. Its direction is read off by conditional expectations from a polynomial soft projector, its small constraint residual is repaired in affine row representations, and one identity accounts for every change of representation. For rational input the algorithm has polynomial bit complexity.
