求解最小封闭球问题的微分私有线性时间fPTAS
A Differentially Private Linear-Time fPTAS for the Minimum Enclosing Ball Problem
摘要：
最小包围球（MEB）问题是聚类中最基本的问题之一，在运筹学、统计学和计算几何中有着广泛的应用。在这项工作中，我们给出了最小封闭球问题的第一个差分私有（DP）fPTAS，改进了Ghazi等人（2020）最知名的DP-PTAS的运行时间和效用界。给定$\R^d$中被球$B(θ_{opt},r_{opt})$覆盖的$n$点，我们的简单迭代DP算法返回一个球$B(θ,r)$，其中$r\leq (1+γ)r_{opt}$，并且在$\tilde O(\nicefrac n {γ^2})$时间中最多留下$\tilde O(\frac{\sqrt d}{γε})$点未被覆盖。我们还给出了我们算法的局部模型版本，该版本最多保留$\tilde O(\frac{\sqrt {nd}}{γε})$点，改进了Nissim和Stemmer（2018）的$n^{0.67}$界（以牺牲其他参数为代价）。此外，我们测试我们的算法经验，并讨论未来的开放问题。
英文摘要：
The Minimum Enclosing Ball (MEB) problem is one of the most fundamental problems in clustering, with applications in operations research, statistics and computational geometry. In this works, we give the first differentially private (DP) fPTAS for the Minimum Enclosing Ball problem, improving both on the runtime and the utility bound of the best known DP-PTAS for the problem, of Ghazi et al. (2020). Given $n$ points in $\R^d$ that are covered by the ball $B(θ_{opt},r_{opt})$, our simple iterative DP-algorithm returns a ball $B(θ,r)$ where $r\leq (1+γ)r_{opt}$ and which leaves at most $\tilde O(\frac{\sqrt d}{γε})$ points uncovered in $\tilde O(\nicefrac n {γ^2})$-time. We also give a local-model version of our algorithm, that leaves at most $\tilde O(\frac{\sqrt {nd}}{γε})$ points uncovered, improving on the $n^{0.67}$-bound of Nissim and Stemmer (2018) (at the expense of other parameters). In addition, we test our algorithm empirically and discuss future open problems.
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