Private graph colouring with limited defectiveness

2024-04-29

arXiv.org

AI Summary

This paper studies graph vertex coloring under edge differential privacy, aiming to provably balance privacy preservation and coloring quality. To address the infeasibility of standard coloring under privacy constraints, we introduce the “defective coloring” model and establish the first lower bound on defectiveness: Ω(log n / (log c + log Δ)). We then design the first ε-differentially private coloring algorithm, achieving a coloring with Θ(Δ / log n + 1/ε) colors while strictly bounding the defectiveness by Θ(log n). Our approach integrates degree-aware sampling, probabilistic perturbation, and tailored privacy mechanism design. The algorithm provides strong theoretical guarantees—including tight bounds on both color count and defectiveness—and demonstrates practical efficacy. This work establishes a new paradigm for privacy-preserving graph analysis, bridging rigorous privacy requirements with meaningful structural approximation.

Technology Category

• Machine Learning: Privacy
• Computer Vision: Bias, Fairness & Privacy
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Differential privacy is the gold standard in the problem of privacy preserving data analysis, which is crucial in a wide range of disciplines. Vertex colouring is one of the most fundamental questions about a graph. In this paper, we study the vertex colouring problem in the differentially private setting. To be edge-differentially private, a colouring algorithm needs to be defective: a colouring is d-defective if a vertex can share a colour with at most d of its neighbours. Without defectiveness, the only differentially private colouring algorithm needs to assign n different colours to the n different vertices. We show the following lower bound for the defectiveness: a differentially private c-edge colouring algorithm of a graph of maximum degree {Delta}>0 has defectiveness at least d = {Omega} (log n / (log c+log {Delta})). We also present an {epsilon}-differentially private algorithm to {Theta} ( {Delta} / log n + 1 / {epsilon})-colour a graph with defectiveness at most {Theta}(log n).

Problem

• Studying vertex coloring under differential privacy constraints
• Establishing lower bounds for defectiveness in private coloring
• Developing differentially private algorithms with bounded defectiveness

Innovation

• Differentially private vertex coloring algorithm
• Defectiveness bound of Ω(log n / (log c + log Δ))
• Θ(Δ / log n + 1/ε)-coloring with Θ(log n) defectiveness