We study the facility location problem under the constraints imposed by local differential privacy (LDP). Recently, Gupta et al. (2010) and Esencayi et al. (2019) proposed lower and upper bounds for the problem on the central differential privacy (DP) model where a trusted curator first collects all data and processes it. In this paper, we focus on the LDP model, where we protect a client’s participation in the facility location instance. Under the HST metric, we show that there is a non-interactive $\epsilon$-LDP algorithm achieving $O(n^{1/4}/\epsilon^2)$-approximation ratio, where $n$ is the size of the metric. On the negative side, we show a lower bound of $\Omega(n^{1/4}/\sqrt{\epsilon})$ on the approximation ratio for any non-interactive $\epsilon$-LDP algorithm. Thus, our results are tight up to a polynomial factor of $\epsilon$. Moreover, unlike previous results, our results generalize to non-uniform facility costs.

In this paper we study the facility location problem in the model of differential privacy (DP) with uniform facility cost. Specifically, we first show that under the hierarchically well-separated tree (HST) metrics and the super-set output setting that was introduced in Gupta et. al., there is an $\epsilon$-DP algorithm that achieves an $O(\frac{1}{\epsilon})$(expected multiplicative) approximation ratio; this implies an $O(\frac{\log n}{\epsilon})$ approximation ratio for the general metric case, where $n$ is the size of the input metric. These bounds improve the best-known results given by Gupta et. al. In particular, our approximation ratio for HST-metrics is independent of $n$, and the ratio for general metrics is independent of the aspect ratio of the input metric. On the negative side, we show that the approximation ratio of any $\epsilon$-DP algorithm is lower bounded by $\Omega(\frac{1}{\sqrt{\epsilon}})$, even for instances on HST metrics with uniform facility cost, under the super-set output setting. The lower bound shows that the dependence of the approximation ratio for HST metrics on $\epsilon$ can not be removed or greatly improved. Our novel methods and techniques for both the upper and lower bound may find additional applications.