98 research outputs found
Differentially Private Partial Set Cover with Applications to Facility Location
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 -fraction of the elements in
the universe, for some . 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 -center/-supplier with outliers. Like
with the Set Cover problem, no algorithm has been able to give non-trivial
guarantees for -center/-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 -fraction of the
population, for , enables us to circumvent the inherent hardness.
Overall, our work is an important step in tackling and understanding
impossibility results in private combinatorial optimization.Comment: 11 pages, 2 figures. Full version of IJCAI 2023 publicatio
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