Breaking the $n^{1.5}$ Additive Error Barrier for Private and Efficient Graph Sparsification via Private Expander Decomposition

TL;DR

This paper introduces a differentially private, polynomial-time algorithm for graph cut sparsification that surpasses previous additive error barriers, achieving an additive error of approximately n^{1.25} while maintaining multiplicative accuracy.

Contribution

It presents the first efficient private algorithm that breaks the n^{1.5} additive error barrier for graph sparsification using private expander decomposition.

Findings

Achieves additive error of n^{1.25 + o(1)} in private graph sparsification.

Provides a polynomial-time differentially private algorithm for expander decomposition.

Improves the accuracy of private graph cut approximations over previous methods.

Abstract

We study differentially private algorithms for graph cut sparsification, a fundamental problem in algorithms, privacy, and machine learning. While significant progress has been made, the best-known private and efficient cut sparsifiers on $n$-node graphs approximate each cut within $\widetilde{O}(n^{1.5})$ additive error and $1+\gamma$ multiplicative error for any $\gamma > 0$ [Gupta, Roth, Ullman TCC'12]. In contrast, "inefficient" algorithms, i.e., those requiring exponential time, can achieve an $\widetilde{O}(n)$ additive error and $1+\gamma$ multiplicative error [Eli{\'a}{\v{s}}, Kapralov, Kulkarni, Lee SODA'20]. In this work, we break the $n^{1.5}$ additive error barrier for private and efficient cut sparsification. We present an $(\varepsilon,\delta)$-DP polynomial time algorithm that, given a non-negative weighted graph, outputs a private synthetic graph approximating all cuts with multiplicative error $1+\gamma$ and additive error $n^{1.25 + o(1)}$ (ignoring dependencies on $\varepsilon, \delta, \gamma$). At the heart of our approach lies a private algorithm for expander decomposition, a popular and powerful technique in (non-private) graph algorithms.