Expert Persona: Senior Research Scientist in Differential Privacy and Algorithmic Theory.
Abstract
This presentation introduces novel results in differentially private graph approximation, specifically focusing on the Multiway Cut (MWC) and Global $k$-Cut problems under the constraint of edge-level differential privacy (DP). The research addresses a critical gap: achieving the best-known non-private approximation ratios while minimizing additive error caused by privacy mechanisms.
For the Multiway Cut problem, the authors propose an efficient algorithm utilizing a simplex embedding relaxation. By employing a novel "shifting trick"—an analytical technique that uses correction coefficients to transfer edge sensitivity—the team demonstrates that adding noise only between terminals and non-terminals is sufficient to preserve privacy. This results in a $1.3$-approximation ratio with an $O(nk/\epsilon)$ additive error, significantly improving upon previous bounds that scaled with $n^{1.5}$. For the Global $k$-Cut problem, the authors establish a tight lower bound of $\Omega(k \log n)$ for pure DP through a rigorous packing argument involving click-and-bridge constructions. The work concludes by suggesting the applicability of these shifting techniques to broader Semidefinite Programming (SDP) relaxations, such as Max-Cut.
Summary of Differentially Private Graph Approximation
00:00:16 Introduction to Graph Privacy: Graph data structures in industry (e.g., user preferences in music apps) contain sensitive interaction data. The goal is to output a synthetic graph $G'$ that preserves combinatorial properties like min-cuts while maintaining edge-level differential privacy.
00:02:26 Edge-Level vs. Node-Level DP: The research focuses on edge-level privacy, where two graphs are considered neighbors if they differ by exactly one edge weight. This protects the presence or absence of a specific interaction between two nodes.
00:03:46 Cut Approximation Standards: Standard private graph analysis allows for cut queries with an additive error of approximately $n^{1.5}/\epsilon$. This work seeks to determine if specific optimization problems, like Multiway Cut, can achieve lower error rates.
Multiway Cut (MWC): Given a set of $k$ terminals, find the minimum weight set of edges that separates all terminals.
Global $k$-Cut: Find the minimum weight set of edges to split a graph into at least $k$ connected components (no fixed terminals).
00:08:41 Multiway Cut Contribution: The speaker presents an algorithm achieving a $\approx 1.3$ multiplicative approximation (matching the best non-private ratio) with an $O(nk/\epsilon)$ additive error. This is shown to be optimal for efficient algorithms as pure DP requires at least $\Omega(n \log k)$ additive error.
00:10:30 The "Shifting Trick" Mechanism: Instead of adding noise to every potential edge in a complete graph (which introduces $O(n^2)$ noise), the algorithm only perturbs edges between terminals and non-terminals. This reduces the total noise added to the system.
00:12:30 Simplex Embedding & LP Relaxation: MWC is relaxed into a Linear Program (LP) using simplex embedding. The privacy is maintained during the solving of this LP. A rounding scheme is then applied as post-processing to achieve the $1.3$ approximation.
00:16:00 Privacy Proof via Correction Coefficients: To prove privacy despite only adding noise to a subset of edges, the authors use "correction coefficients." This analytical coupling shows that any change in an edge between two non-terminals can be "shifted" or canceled out by adjusting the noise between terminals, rendering the output indistinguishable.
00:27:26 Utility Optimization via the Dual Problem: To improve accuracy, the authors analyze the "uncut" edges (the dual perspective). Because the noise is dense in the cut but sparse in the uncut edges, maximizing the width of edges kept in the graph results in better utility than directly minimizing the cut.
00:29:11 Global $k$-Cut Lower Bounds: The research establishes a lower bound of $k \log n$ for the Global $k$-Cut problem. This indicates that the privacy cost for $k$-Cut is fundamentally different (logarithmic in $n$) compared to MWC (linear in $n$).
00:31:07 Packing Argument Construction: The lower bound is proven by constructing a "hard case" graph consisting of multiple cliques connected by thin "bridges." By showing that an accurate algorithm must distinguish between which bridges were removed, they demonstrate a contradiction with the requirements of DP.
00:41:01 Future Research Directions: The authors aim to extend the "shifting trick" to other combinatorial problems that can be formulated as Semidefinite Programs (SDPs), specifically Max-Cut, to achieve high-fidelity private approximations.