Differentially Private Partial Set Cover with Applications to Facility Location

TL;DR

This paper introduces differentially private algorithms for Partial Set Cover, overcoming previous hardness results, and applies these to develop private approximation algorithms for facility location problems with outliers.

Contribution

It presents the first differentially private algorithm for explicit Partial Set Cover and extends this to private approximation algorithms for facility location problems with relaxed coverage.

Findings

First differentially private explicit Partial Set Cover algorithm.

Achieves non-trivial approximation guarantees under differential privacy.

Enables private solutions for facility location problems with outliers.

Abstract

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.