近似输入稀疏时间中的Spencer定理
Spencer's theorem in nearly input-sparsity time
摘要：
Spencer的一个著名定理指出，对于任何集合系统$S_1,\dots, S_m \subseteq [n]$，都有一个具有$\{\pm 1\}$的基集的着色，该$\{\pm 1\}$具有差异$O(\sqrt{n\log(m/n+2)})$。我们提供了一个算法来找到这样的着色在近输入稀疏时间$\tilde{O}(n+\sum_{i=1}^{m}|S_i|)$。在我们的工作中，这可能是独立的利益，是一种新的宽度减少技术解决线性规划，而不是覆盖/包装类型，在近输入稀疏时间使用乘法权重更新方法。
英文摘要：
A celebrated theorem of Spencer states that for every set system $S_1,\dots, S_m \subseteq [n]$, there is a coloring of the ground set with $\{\pm 1\}$ with discrepancy $O(\sqrt{n\log(m/n+2)})$. We provide an algorithm to find such a coloring in near input-sparsity time $\tilde{O}(n+\sum_{i=1}^{m}|S_i|)$. A key ingredient in our work, which may be of independent interest, is a novel width reduction technique for solving linear programs, not of covering/packing type, in near input-sparsity time using the multiplicative weights update method.
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