We provide an approximation algorithm for k-means clustering in the one-round (aka non-interactive) local model of differential privacy (DP). This algorithm achieves an approximation ratio arbitrarily close to the best non private approximation algorithm, improving upon previously known algorithms that only guarantee large (constant) approximation ratios. Furthermore, this is the first constant-factor approximation algorithm for k-means that requires only one round of communication in the local DP model, positively resolving an open question of Stemmer (SODA 2020). Our algorithmic framework is quite flexible; we demonstrate this by showing that it also yields a similar near-optimal approximation algorithm in the (one-round) shuffle DP model.

翻译：我们为一回合(不互动的)当地差异隐私模式中的 k 手段分组提供了一种近似算法。 这个算法实现了任意接近于最佳非私人近似算法的近似比率,改进了以前已知的仅保证大( 固定的) 近似比率的算法。 此外, 这是K 手段的第一个恒定因素近似算法, 只需要本地DP 模式的一回合通信, 就能正面解决 Stemmer (SODA 2020) 的未决问题 。 我们的算法框架相当灵活; 我们通过显示它在( 一回合) 的 DP 模式中也产生类似的近乎最佳的近似近似算法来证明这一点 。