The Price of Privacy For Approximating Max-CSP
Prathamesh Dharangutte, Jingcheng Liu, Pasin Manurangsi, Akbar Rafiey, Phanu Vajanopath, Zongrui Zou
TL;DR
This work initiates a systematic study of approximating Max-CSPs under differential privacy when constraints are private. It identifies a high-privacy regime where no DP algorithm can outperform a random assignment by more than a factor of $O( ext{$\varepsilon$})$, and provides a polynomial-time algorithm achieving this bound on triangle-free bounded-degree instances, with extensions to Max-Cut and Max-$k$XOR under various assumptions. In the low-privacy regime, the paper shows near-optimal performance via the Exponential Mechanism, yielding additive deficits and multiplicative bounds that depend on $ ext{$\varepsilon$}$ and problem size, and provides matching lower bounds. A rich set of techniques—private partitioning, privatized versions of Shearer’s Max-Cut algorithm, degree-based decompositions, and center-polynomial analyses—enable private improvements over random for several CSPs, including odd $k$ Max-$k$XOR and Max-Cut, even when triangle-freeness or bounded degree are relaxed at the cost of efficiency. The results illuminate fundamental privacy-utility tradeoffs in DP Max-CSPs and offer concrete algorithms with provable privacy and utility guarantees, highlighting both near-optimal regimes and open avenues for broader generalization and efficiency gains.
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.