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We study the Densest Subgraph (DSG) problem under the additional constraintof differential privacy. DSG is a fundamental theoretical question which playsa central role in graph analytics, and so privacy is a natural requirement. Allknown private algorithms for Densest Subgraph lose constant multiplicativefactors, despite the existence of non-private exact algorithms. We show that,perhaps surprisingly, this loss is not necessary: in both the classicdifferential privacy model and the LEDP model (local edge differential privacy,introduced recently by Dhulipala et al. [FOCS 2022]), we give $(\epsilon,\delta)$-differentially private algorithms with no multiplicative losswhatsoever. In other words, the loss is \emph{purely additive}. Moreover, ouradditive losses match or improve the best-known previous additive loss (in anyversion of differential privacy) when $1/\delta$ is polynomial in $n$, and arealmost tight: in the centralized setting, our additive loss is $O(\log n/\epsilon)$ while there is a known lower bound of $\Omega(\sqrt{\log n /\epsilon})$.  We also give a number of extensions. First, we show how to extend ourtechniques to both the node-weighted and the directed versions of the problem.Second, we give a separate algorithm with pure differential privacy (as opposedto approximate DP) but with worse approximation bounds. And third, we give anew algorithm for privately computing the optimal density which implies aseparation between the structural problem of privately computing the densestsubgraph and the numeric problem of privately computing the density of thedensest subgraph.

Almost Tight Bounds for Differentially Private Densest Subgraph