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

2025-07-02

AI Summary

This work addresses the design of efficient differentially private graph sparsification algorithms for cut approximation. Existing efficient algorithms incur an additive error of $widetilde{O}(n^{1.5})$ and a multiplicative factor of $1+gamma$ for arbitrary cuts, whereas optimal—but computationally infeasible—algorithms achieve $widetilde{O}(n)$ additive error. We present the first polynomial-time algorithm that breaks the $n^{1.5}$ additive error barrier, achieving joint guarantees of $widetilde{O}(n^{1.25+o(1)})$ additive error and $1+gamma$ multiplicative error. Our core technique integrates differentially private expander decomposition with synthetic graph generation to construct high-fidelity private graph representations. The algorithm applies to nonnegative-weighted graphs and provides uniform approximation for all cuts. This significantly improves the accuracy–efficiency trade-off for cut queries on private graphs, advancing the state of the art in differentially private graph analytics.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Privacy-enhancing technologies
• Responsible Web: Data and user privacy-enhancing technologies for the Web

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 $1+γ$ multiplicative error for any $γ> 0$ [Gupta, Roth, Ullman TCC'12]. In contrast, "inefficient" algorithms, i.e., those requiring exponential time, can achieve an $widetilde{O}(n)$ additive error and $1+γ$ multiplicative error [Eli{á}{š}, 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 $(varepsilon,δ)$-DP polynomial time algorithm that, given a non-negative weighted graph, outputs a private synthetic graph approximating all cuts with multiplicative error $1+γ$ and additive error $n^{1.25 + o(1)}$ (ignoring dependencies on $varepsilon, δ, γ$). At the heart of our approach lies a private algorithm for expander decomposition, a popular and powerful technique in (non-private) graph algorithms.

Problem

• Improving additive error in private graph sparsification
• Achieving efficient private cut approximation
• Developing private expander decomposition techniques

Innovation

• Private expander decomposition for graph sparsification
• Polynomial time algorithm with improved additive error
• Differentially private synthetic graph generation