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. [13]. The current best known expected approximation ratio for an ϵ-DP algorithm is $O\left(\frac{\log n}{\sqrt{ϵ}}\right)$ due to Cohen-Addad et al. [3] where n denote the size of the metric space, meanwhile the best known lower bound is $\Omega (1/\sqrt{ϵ})$ [8].

In this short note, we give a lower bound of $\tilde{\Omega }(\min \{\log n,\sqrt{\frac{\log n}{ϵ}}\})$ 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.