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 $(\epsilon, \delta)$-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 $1/\delta$ is polynomial in $n$, and are almost 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 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.

翻译：我们研究在差分隐私额外约束下的稠密子图问题。稠密子图是一个基础理论问题，在图分析中扮演核心角色，因此隐私保护是自然需求。尽管存在非私有的精确算法，所有已知的稠密子图私有算法都会损失常数乘法因子。我们证明，这种损失并非必要：在经典差分隐私模型和LEDP模型（局部边差分隐私，由Dhulipala等人最近提出于FOCS 2022）中，我们给出了完全不损失任何乘法因子的$(\epsilon, \delta)$-差分隐私算法。换言之，损失是纯粹的加法性。此外，当$1/\delta$关于$n$为多项式时，我们的加法损失匹配或改进了先前已知的最佳加法损失（在任何版本的差分隐私中），并且几乎是严格的：在集中式设置中，我们的加法损失为$O(\log n /\epsilon)$，而已知下界为$\Omega(\sqrt{\log n / \epsilon})$。我们还给出了若干扩展。首先，我们展示了如何将我们的技术扩展至问题的节点加权版本和有向版本。其次，我们给出了一个具有纯差分隐私（而非近似差分隐私）但近似界较差的独立算法。第三，我们提出了一个用于私有计算最优密度的新算法，该算法在私有计算稠密子图的结构性问题与私有计算稠密子图密度的数值问题之间实现了分离。