# On Facility Location Problem in the Local Differential Privacy Model

25th International Conference on Artificial Intelligence and Statistics (AISTATS 2022) (2022)

## Abstract

We study the uncapacitated facility location (UFL) problem under the constraints imposed by the local differential privacy (LDP). Recently, Gupta et al. (2009) and Esencayi et al. (2019) proposed lower and upper bounds for the UFL problem on the central differential privacy (DP) model where a curator first collects all data before being processed. 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 factor of $\epsilon$. Moreover, unlike previous results, our results generalize for non-uniform facility costs.