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 $(\epsilon, \delta)$-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 $1/\delta$ is at least 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 $\Omega(\sqrt{\log n / \epsilon})$. Additionally, we give a different algorithm that is $\epsilon$-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.

翻译：我们研究了在差分隐私附加约束下的稠密子图（DSG）问题。DSG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。但所有已知的稠密子图隐私算法在不存在非私密精确算法的情况下，都会损失常数倍因子以及相对较大的（至少$\log^2 n$）加性因子。我们证明，或许令人惊讶的是，这些损失并非必然：在经典差分隐私模型和LEDP模型（最近由Dhulipala等人[FOCS 2022]提出的本地边差分隐私）中，我们给出了没有任何倍数损失的$(\epsilon, \delta)$-差分隐私算法。换言之，损失纯粹是加性的。此外，当$1/\delta$至少是$n$的多项式时，我们的加性损失改进了先前已知的最佳加性损失（在任何版本的差分隐私中），并且几乎紧：在中心化设置中，我们的加性损失为$O(\log n /\epsilon)$，而已知下界为$\Omega(\sqrt{\log n / \epsilon})$。另外，我们提出了一种在LEDP模型中满足$\epsilon$-差分隐私的不同算法，该算法实现了任意接近$2$的倍数比率，并带有额外的加性因子。这改进了LEDP模型中先前倍数$4$逼近的结果。最后，我们总结了将我们的技术扩展到问题的节点加权版本和有向版本的应用。