Fine-Grained Privacy Guarantees for Coverage Problems

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.