The Price of Privacy For Approximating Max-CSP

2026-02-09

AI Summary

This work investigates the approximate solvability of the maximum constraint satisfaction problem (Max-CSP) under differential privacy constraints, where the constraints constitute sensitive data. Focusing on the high-privacy regime (ε ≪ 1), the study combines differential privacy mechanisms, approximation algorithm analysis, and structural graph assumptions—such as bounded degree and triangle-freeness—to establish that any ε-differentially private algorithm can achieve an approximation ratio at most O(ε) better than random assignment. A polynomial-time algorithm matching this upper bound is designed. Furthermore, tight approximation guarantees are achieved for instances with bounded degree and no triangles, and the structural requirements are relaxed for specific problems like Max-Cut, thereby extending the theoretical limits of efficient private approximation in the high-privacy setting.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Privacy
• Reasoning under Uncertainty: Stochastic 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: Privacy-enhancing technologies
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We study approximation algorithms for Maximum Constraint Satisfaction Problems (Max-CSPs) under differential privacy (DP) where the constraints are considered sensitive data. Information-theoretically, we aim to classify the best approximation ratios possible for a given privacy budget $\varepsilon$. In the high-privacy regime ($\varepsilon \ll 1$), we show that any $\varepsilon$-DP algorithm cannot beat a random assignment by more than $O(\varepsilon)$ in the approximation ratio. We devise a polynomial-time algorithm which matches this barrier under the assumptions that the instances are bounded-degree and triangle-free. Finally, we show that one or both of these assumptions can be removed for specific CSPs--such as Max-Cut or Max $k$-XOR--albeit at the cost of computational efficiency.

Problem

• differential privacy
• Max-CSP
• approximation ratio
• privacy budget
• constraint satisfaction problems

Innovation

• differential privacy
• Max-CSP
• approximation algorithms
• privacy-accuracy tradeoff
• bounded-degree graphs