Almost Tight Bounds for Differentially Private Densest Subgraph

TL;DR

This work studies the Densest Subgraph problem under differential privacy, addressing whether one can avoid multiplicative privacy loss. The authors develop a novel private Hedge/MWU framework embedded in the PST LP-solving scheme to obtain purely additive private DSG algorithms in both centralized DP and Local Edge DP (LEDP) models, with additive errors that scale polylogarithmically in the graph size. They extend the core approach to node-weighted and directed variants, provide a simple ε-LEDP algorithm with favorable round complexity, and give a private density-approximation strategy that separates the numeric value from the private structure. Centralized improvements via Papernot–Steinke further tighten the additive guarantees. Together, these results significantly advance private graph analytics by achieving additive-only losses and near-optimal accuracy for DSG under multiple privacy models, with practical extensions and concrete privacy/utility tradeoffs.

Abstract

We study the Densest Subgraph (DSG) problem under the additional constraint of differential privacy. DSG is a fundamental theoretical question which plays a central role in graph analytics, and so privacy is a natural requirement. All known private algorithms for Densest Subgraph lose constant multiplicative factors, despite the existence of non-private exact algorithms. We show that, perhaps surprisingly, this loss is not necessary: in both the classic differential privacy model and the LEDP model (local edge differential privacy, introduced recently by Dhulipala et al. [FOCS 2022]), we give $(ε, δ)$-differentially private algorithms with no multiplicative loss whatsoever. In other words, the loss is \emph{purely additive}. Moreover, our additive losses match or improve the best-known previous additive loss (in any version of differential privacy) when $1/δ$ is polynomial in $n$, and are almost tight: in the centralized setting, our additive loss is $O(\log n /ε)$ while there is a known lower bound of $Ω(\sqrt{\log n / ε})$. We also give a number of extensions. First, we show how to extend our techniques to both the node-weighted and the directed versions of the problem. Second, we give a separate algorithm with pure differential privacy (as opposed to approximate DP) but with worse approximation bounds. And third, we give a new algorithm for privately computing the optimal density which implies a separation between the structural problem of privately computing the densest subgraph and the numeric problem of privately computing the density of the densest subgraph.

Theorems & Definitions (112)

• Definition 1
• Definition 2: Edge-Neighboring NRS07
• Definition 3: Edge Differential Privacy NRS07
• Definition 4: Zero-Concentrated Differential Privacy(zCDP) bun2016concentrated
• Theorem 1
• Theorem 2
• proof
• Lemma 3: Lemma 4.2 of CQT22
• Lemma 4
• proof