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 linear time 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 . 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 Õ((equation presented)) points uncovered in Õ(n/γ2)-time. We also give a local-model version of our algorithm, that leaves at most Õ((equation presented)) points uncovered, improving on the n0.67-bound of Nissim and Stemmer  (at the expense of other parameters). Lastly, we test our algorithm empirically and discuss open problems.
|Title of host publication||Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022|
|Editors||S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, A. Oh|
|Publisher||Neural information processing systems foundation|
|State||Published - 2022|
|Event||36th Conference on Neural Information Processing Systems, NeurIPS 2022 - New Orleans, United States|
Duration: 28 Nov 2022 → 9 Dec 2022
|Name||Advances in Neural Information Processing Systems|
|Conference||36th Conference on Neural Information Processing Systems, NeurIPS 2022|
|Period||28/11/22 → 9/12/22|
Bibliographical noteFunding Information:
O.S. is supported by the BIU Center for Research in Applied Cryptography and Cyber Security in conjunction with the Israel National Cyber Bureau in the Prime Ministers Office, and by ISF grant no. 2559/20. Both authors thank the anonymous reviewers for terrific suggestions and advice on improving this paper.
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