It was observed in \citet{gupta2009differentially} that the Set Cover problem has strong impossibility results under differential privacy. In our work, we observe that these hardness results dissolve when we turn to the Partial Set Cover problem, where we only need to cover a $\rho$-fraction of the elements in the universe, for some $\rho\in(0,1)$. We show that this relaxation enables us to avoid the impossibility results: under loose conditions on the input set system, we give differentially private algorithms which output an explicit set cover with non-trivial approximation guarantees. In particular, this is the first differentially private algorithm which outputs an explicit set cover. Using our algorithm for Partial Set Cover as a subroutine, we give a differentially private (bicriteria) approximation algorithm for a facility location problem which generalizes $k$-center/$k$-supplier with outliers. Like with the Set Cover problem, no algorithm has been able to give non-trivial guarantees for $k$-center/$k$-supplier-type facility location problems due to the high sensitivity and impossibility results. Our algorithm shows that relaxing the covering requirement to serving only a $\rho$-fraction of the population, for $\rho\in(0,1)$, enables us to circumvent the inherent hardness. Overall, our work is an important step in tackling and understanding impossibility results in private combinatorial optimization.

翻译：在\citet{gupta2009differentially}的研究中观察到，集合覆盖问题在差分隐私条件下存在强不可能性结果。我们的工作发现，当转向部分集合覆盖问题时，这些困难结果得以消解——部分集合覆盖问题只需覆盖全域中$\rho$比例的元素（$\rho\in(0,1)$）。我们证明这种松弛能够避免不可能性结果：在输入集合系统的宽松条件下，我们给出了具有非平凡近似保证的差分隐私算法，该算法能输出显式集合覆盖。值得注意的是，这是首个输出显式集合覆盖的差分隐私算法。通过将部分集合覆盖算法作为子程序，我们为含离群点的$k$-中心/$k$-供应商类设施选址问题提出了一个差分隐私（双准则）近似算法。与集合覆盖问题类似，由于高敏感性和不可能性结果，此前没有任何算法能对$k$-中心/$k$-供应商类设施选址问题给出非平凡保证。我们的算法表明，将覆盖要求松弛为仅服务$\rho$比例的人口（$\rho\in(0,1)$），能够规避固有的困难性。总体而言，我们的工作在解决和理解私有组合优化中的不可能性结果方面迈出了重要一步。