We introduce a new notion of neighboring databases for coverage problems such as Max Cover and Set Cover under differential privacy. In contrast to the standard privacy notion for these problems, which is analogous to node-privacy in graphs, our new definition gives a more fine-grained privacy guarantee, which is analogous to edge-privacy. We illustrate several scenarios of Set Cover and Max Cover where our privacy notion is desired one for the application. Our main result is an $\epsilon$-edge differentially private algorithm for Max Cover which obtains an $(1-1/e-\eta,\tilde{O}(k/\epsilon))$-approximation with high probability. Furthermore, we show that this result is nearly tight: we give a lower bound show that an additive error of $\Omega(k/\epsilon)$ is necessary under edge-differential privacy. Via group privacy properties, this implies a new algorithm for $\epsilon$-node differentially private Max Cover which obtains an $(1-1/e-\eta,\tilde{O}(fk/\epsilon))$-approximation, where $f$ is the maximum degree of an element in the set system. When $f\ll k$, this improves over the best known algorithm for Max Cover under pure (node) differential privacy, which obtains an $(1-1/e,\tilde{O}(k^2/\epsilon))$-approximation.

翻译：我们针对差分隐私下的最大覆盖和集合覆盖等覆盖问题，引入了一种新的相邻数据库定义。与这些问题中的标准隐私概念（类似于图中的节点隐私）不同，我们的新定义提供了更细粒度的隐私保证，类似于边隐私。我们阐述了集合覆盖和最大覆盖的若干场景，在这些场景中，我们的隐私概念是应用所期望的。我们的主要成果是为最大覆盖问题设计了一个$\epsilon$-边差分隐私算法，该算法能以高概率获得$(1-1/e-\eta,\tilde{O}(k/\epsilon))$近似比。此外，我们证明了该结果几乎是紧的：我们给出了一个下界，表明在边差分隐私下，$\Omega(k/\epsilon)$的加法误差是必要的。通过群组隐私性质，这为$\epsilon$-节点差分隐私下的最大覆盖问题提供了一种新算法，该算法获得$(1-1/e-\eta,\tilde{O}(fk/\epsilon))$近似比，其中$f$是集合系统中元素的最大度数。当$f\ll k$时，这改进了纯（节点）差分隐私下已知的最佳最大覆盖算法，后者获得$(1-1/e,\tilde{O}(k^2/\epsilon))$近似比。