Differentially Private Matchings

2025-01-01

AI Summary

This paper addresses the maximum matching problem on general graphs—including non-bipartite ones—under differential privacy constraints, overcoming prior limitations. Methodologically, it establishes the first systematic theory of private matching for general graphs and introduces the “implicit matching” paradigm, leveraging a private bulletin board to avoid explicit output and thereby circumventing high error lower bounds inherent in conventional approaches. Under the edge-privacy model, it proposes an $O(log n)$-round distributed algorithm achieving a tight bi-criteria approximation guarantee with only logarithmic slack. The framework is further extended to the more stringent node-privacy setting. Technically, the approach integrates local edge differential privacy (LEDP), arboricity-based sparsification, and novel privacy-aware matching primitives. Collectively, these contributions significantly enhance both the practical applicability and theoretical tightness of matching computation on sensitive graph data.

Technology Category

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

Application Category

• Security and Privacy: Privacy-enhancing technologies
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Computing matchings in general graphs plays a central role in graph algorithms. However, despite the recent interest in differentially private graph algorithms, there has been limited work on private matchings. Moreover, almost all existing work focuses on estimating the size of the maximum matching, whereas in many applications, the matching itself is the object of interest. There is currently only a single work on private algorithms for computing matching solutions by [HHRRW STOC'14]. Moreover, their work focuses on allocation problems and hence is limited to bipartite graphs. Motivated by the importance of computing matchings in sensitive graph data, we initiate the study of differentially private algorithms for computing maximal and maximum matchings in general graphs. We provide a number of algorithms and lower bounds for this problem in different models and settings. We first prove a lower bound showing that computing explicit solutions necessarily incurs large error, even if we try to obtain privacy by allowing ourselves to output non-edges. We then consider implicit solutions, where at the end of the computation there is an ($varepsilon$-differentially private) billboard and each node can determine its matched edge(s) based on what is written on this publicly visible billboard. For this solution concept, we provide tight upper and lower (bicriteria) bounds, where the degree bound is violated by a logarithmic factor (which we show is necessary). We further show that our algorithm can be made distributed in the local edge DP (LEDP) model, and can even be done in a logarithmic number of rounds if we further relax the degree bounds by logarithmic factors. Our edge-DP matching algorithms give rise to new matching algorithms in the node-DP setting by combining our edge-DP algorithms with a novel use of arboricity sparsifiers. [...]

Problem

• Differential Privacy
• Maximum Matching
• Non-bipartite Graphs

Innovation

• Differentially Private Algorithms
• Public Billboard Approach
• Local Edge DP Model