We give a concentration inequality for a stochastic version of the facility location problem. We show the objective $C_n = \min_{F \subseteq [0,1]^2}|F|+\sum_{x\in X}\min_{f\in F}\|x-f\|$ is concentrated in an interval of length $O(n^{1/6})$ and $\E[C_n]=\Theta(n^{2/3})$ if the input $X$ consists of i.i.d. uniform points in the unit square. Our main tool is to use a geometric quantity, previously used in the design of approximation algorithms for the facility location problem, to analyze a martingale process. Many of our techniques generalize to other settings.
翻译:我们给设施位置问题的一个随机版本给出了浓度不平等。 我们给出了 $C_n =\ min ⁇ F\ subseteq [0, 1,2 ⁇ F sup ⁇ x\xèin X ⁇ min ⁇ f\ in F ⁇ x- f ⁇ }$(n ⁇ 1/6}) 和$\ E[C_n] {Theta(n ⁇ 2/3}) 之间的时间间隔。 如果输入的 $X$ 包含单位方形的 i. d. 统一点, 我们的主要工具是使用几何数数量, 先前用于设计设施位置问题的近似算法, 分析马丁格过程。 我们的许多技术都概括到其他设置 。