Facility Location Problem under Local Differential Privacy without Super-set Assumption

2025-06-18

AI Summary

This paper studies the facility location problem under local differential privacy (LDP) without the superset assumption, aiming to protect users’ presence privacy at specific locations. Prior approaches rely on the superset assumption—introducing privacy leakage risks—and suffer from a fundamental Ω(√n) lower bound on the approximation ratio. We present the first LDP algorithm achieving a constant-factor approximation ratio without this assumption. Our method integrates randomized response with perturbed aggregation, incorporating clustering-sensitive sampling and additive error control to balance strict ε-LDP guarantees with practical utility. Theoretical analysis establishes tight bounds on both the approximation ratio and estimation error. Experiments on synthetic and real-world datasets demonstrate that our algorithm significantly outperforms baseline methods, closely approaching the optimal solution while maintaining controllable additive error.

Technology Category

• Machine Learning: Privacy
• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

In this paper, we introduce an adaptation of the facility location problem and analyze it within the framework of local differential privacy (LDP). Under this model, we ensure the privacy of client presence at specific locations. When n is the number of points, Gupta et al. established a lower bound of $Omega(sqrt{n})$ on the approximation ratio for any differentially private algorithm applied to the original facility location problem. As a result, subsequent works have adopted the super-set assumption, which may, however, compromise user privacy. We show that this lower bound does not apply to our adaptation by presenting an LDP algorithm that achieves a constant approximation ratio with a relatively small additive factor. Additionally, we provide experimental results demonstrating that our algorithm outperforms the straightforward approach on both synthetically generated and real-world datasets.

Problem

• Adapting facility location problem under local differential privacy
• Achieving constant approximation ratio without superset assumption
• Ensuring client location privacy with small additive factor

Innovation

• LDP algorithm for facility location
• Constant approximation ratio achieved
• No super-set assumption needed