Location problems with privacy

We examine well known facility location problems under the privacy challenges posed by big data environments. For a given set of n points U∈R^d, previous works have introduced the “Topology Descriptor Grid” (TDG) [1,2], a privacy-preserving framework under which some approximate solutions are possible for a variety of clustering problems. In this paper, we introduce the Equidistant “Location Estimation using Concentric Circles” (LECC) framework in R², which obfuscates exact point locations while preserving their relative distances to a predetermined point. We show, under this new framework, how to obtain 2+O(1/n)-approximate solutions for the 1-center, 1-median, 1-mean, and k-centrum problems, and O(k),O(k),O(k²) approximations for the k-center, k-median and k-means problems, respectively. For the TDG framework we provide a (d,k^d−1),(d,k^d−1), and (d²,k^d−1) approximations for the k-center, k-median, and k-means problems, respectively. Additionally, for the single facility ordered median problem, if every two weights of any two points are upper bounded by α, the ℓ₁ ordered median constitutes a α²dd-approximation for the ℓ₂ ordered median, regardless of any privacy-preserving framework.