The Price of Privacy For Approximating Max-CSP

TL;DR

This paper investigates the limits of approximation algorithms for Max-CSPs under differential privacy, establishing theoretical bounds and providing algorithms that match these bounds under certain conditions.

Contribution

It characterizes the best possible approximation ratios under differential privacy and offers polynomial-time algorithms that achieve these bounds for specific Max-CSPs.

Findings

Any ε-DP algorithm cannot outperform random assignment by more than O(ε) in high-privacy regimes.

A polynomial-time algorithm matches the theoretical privacy-bound approximation ratio for bounded-degree, triangle-free instances.

For Max-Cut and Max k-XOR, assumptions can be relaxed at the expense of computational efficiency.

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.