A Differentially Private Linear-Time fPTAS for the Minimum Enclosing Ball Problem

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 Rd that are covered by the ball B(θopt,ropt), our simple iterative DP-algorithm returns a ball B(θ,r) where r≤ (1+γ)ropt and which leaves at most tilde O(frac√ dγε) points uncovered in tilde O(nicefrac n γ2)-time. We also give a local-model version of our algorithm, that leaves at most tilde O(frac√ ndγε) points uncovered, improving on the n0.67-bound of Nissim and Stemmer (2018) (at the expense of other parameters). In addition, we test our algorithm empirically and discuss future open problems.