We study the Densest Subgraph problem under the additional constraint of differential privacy. In the LEDP (local edge differential privacy) model, introduced recently by Dhulipala et al. [FOCS 2022], we give an $(\epsilon, \delta)$-differentially private algorithm with no multiplicative loss: the loss is purely additive. This is in contrast to every previous private algorithm for densest subgraph (local or centralized), all of which incur some multiplicative loss as well as some additive loss. Moreover, our additive loss matches the best-known previous additive loss (in any version of differential privacy) when $1/\delta$ is at least polynomial in $n$, and in the centralized setting we can strengthen our result to provide better than the best-known additive loss. 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额外约束下的稠密子图问题。在最近由Dhulipala等人[FOCS 2022]引入的LEDP(局部边差分隐私)模型中，我们给出了一种$(\epsilon, \delta)$-差分隐私算法，其无乘法损失:损失纯属加法性。这与以往所有稠密子图隐私算法(无论是局部还是集中式)形成鲜明对比，这些算法都会同时产生乘法损失和加法损失。此外，当$1/\delta$在$n$的多项式量级时，我们的加法损失与已知最佳加法损失(在任何差分隐私版本中)相匹配，且在集中式设置中，我们可以强化结果，使其优于已知最佳加法损失。同时，我们给出另一种在LEDP模型中实现$\epsilon$-差分隐私的算法，该算法在附加加法因子的前提下，可实现任意接近2的乘法比。这改进了LEDP模型之前乘法4-近似的性能。最后，我们将技术扩展至节点加权版本和有向版本的问题。