We study the task of differentially private clustering. For several basic
clustering problems, including Euclidean DensestBall, 1-Cluster, k-means, and
k-median, we give efficient differentially private algorithms that achieve
essentially the same approximation ratios as those that can be obtained by any
non-private algorithm, while incurring only small additive errors. This
improves upon existing efficient algorithms that only achieve some large
constant approximation factors.
Our results also imply an improved algorithm for the Sample and Aggregate
privacy framework. Furthermore, we show that one of the tools used in our
1-Cluster algorithm can be employed to get a faster quantum algorithm for
ClosestPair in a moderate number of dimensions.

本论文研究了不同 ially private clustering 任务，为 Euclidean DensestBall、1-Cluster、k-means 和 k-median 等基本聚类问题提供了有效的差分隐私算法，同时只产生小的附加误差，从而实现了与任何非私有算法可以获得的近似比例基本相同的近似比例。这改进了现有的仅实现某些大常数逼近因子的有效算法。我们的结果还暗示了改进的 Sample and Aggregate 隐私框架算法。此外，我们展示了在适度的维数下，可以利用我们的 1-Cluster 算法中使用的工具来获得更快的 ClosestPair 量子算法。