Approximation algorithms for facility location and k-median with differential privacy

Abstract

In this paper we consider the problems of facility location and k-median with differential privacy in metric space, where a local search-based framework is proposed to solve the differential privacy issues. The approximation algorithm given for the facility location problem has a multiplicative error of 4 and an additive error of $O(\Delta n^2\log n\log (n+f_{\max }{\Delta }^{-1}\left){\epsilon }^{-1}\right)$, where $f_{\max }$ is the maximum facility-opening cost, n is the number of clients, and Δ is the maximum distance between any two input points. For the k-median problem, our local search-based framework yields an approximation algorithm with a multiplicative error of $4+\epsilon$ and an additive error of $O(\Delta k^2{\log }^{2}n{\epsilon }^{-2})$.