Improved Lower Bound for Differentially Private Facility Location

We consider the differentially private (DP) facility location problem in the so called super-set output setting proposed by Gupta et al. [SODA 2010]. The current best known expected approximation ratio for an ϵ-DP algorithm is O(log n/√(ϵ)) due to Cohen-Addad et al. [AISTATS 2022] where n denote the size of the metric space, meanwhile the best known lower bound is Ω(1/√(ϵ)) [NeurIPS 2019]. In this short note, we give a lower bound of Ω̃(min{log n, √(log n/ϵ)}) on the expected approximation ratio of any ϵ-DP algorithm, which is the first evidence that the approximation ratio has to grow with the size of the metric space.