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 $\epsilon$-DP algorithm is $O\left(\frac{\log n}{\sqrt{\epsilon}}\right)$ due to Cohen-Addad et al. [AISTATS 2022] where $n$ denote the size of the metric space, meanwhile the best known lower bound is $\Omega(1/\sqrt{\epsilon})$ [NeurIPS 2019]. In this short note, we give a lower bound of $\tilde{\Omega}\left(\min\left\{\log n, \sqrt{\frac{\log n}{\epsilon}}\right\}\right)$ on the expected approximation ratio of any $\epsilon$-DP algorithm, which is the first evidence that the approximation ratio has to grow with the size of the metric space.

翻译：我们考虑Gupta等人[SODA 2010]提出的超集输出设置中的差分隐私设施选址问题。目前，Cohen-Addad等人[AISTATS 2022]设计的$\epsilon$-DP算法取得了已知最佳期望近似比$O\left(\frac{\log n}{\sqrt{\epsilon}}\right)$，其中$n$表示度量空间的规模；而此前已知最佳下界为$\Omega(1/\sqrt{\epsilon})$[NeurIPS 2019]。在本短文中，我们给出了任意$\epsilon$-DP算法期望近似比的下界$\tilde{\Omega}\left(\min\left\{\log n, \sqrt{\frac{\log n}{\epsilon}}\right\}\right)$，这是首次证明该近似比必须随度量空间规模增长。