Almost Tight Bounds for Differentially Private Densest Subgraph

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. But all known private algorithms for Densest Subgraph lose constant multiplicative factors as well as relative large (at least log^2 n) additive factors, despite the existence of non-private exact algorithms. We show that, perhaps surprisingly, these losses are 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 purely additive. Moreover, our additive losses improve the best-known previous additive loss (in any version of differential privacy) when 1/δ is at least 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(√(log n / ϵ)). Additionally, we give a different algorithm that is ϵ-differentially private in the LEDP model which achieves a multiplicative ratio arbitrarily close to 2, along with an additional additive factor. This improves over the previous multiplicative 4-approximation in the LEDP model. Finally, we conclude with extensions of our techniques to both the node-weighted and the directed versions of the problem.