# Differentially Private Approximation Algorithms

journal contribution

posted on 01.08.2009, 00:00 by Anupam Gupta, Katrina Ligett, Frank McSherry, Aaron Roth, Kunal TalwarConsider the following problem: given a metric space, some of whose points are “clients”, open a set
of at most k facilities to minimize the average distance from the clients to these facilities. This is just the
well-studied k-median problem, for which many approximation algorithms and hardness results are known.
Note that the objective function encourages opening facilities in areas where there are many clients, and given
a solution, it is often possible to get a good idea of where the clients are located. However, this poses the
following quandary: what if the identity of the clients is sensitive information that we would like to keep
private? Is it even possible to design good algorithms for this problem that preserve the privacy of the clients?
In this paper, we initiate a systematic study of algorithms for discrete optimization problems in the
framework of differential privacy (which formalizes the idea of protecting the privacy of individual input
elements). We show that many such problems indeed have good approximation algorithms that preserve
differential privacy; this is even in cases where it is impossible to preserve cryptographic definitions of privacy
while computing any non-trivial approximation to even the value of an optimal solution, let alone the entire
solution.
Apart from the k-median problem, we consider the problems of vertex and set cover, min-cut, k-median,
facility location, and Steiner tree, and give approximation algorithms and lower bounds for these problems.
We also consider the recently introduced submodular maximization problem, “Combinatorial Public Projects”
(CPP), shown by Papadimitriou et al. [PSS08] to be inapproximable to subpolynomial multiplicative factors
by any efficient and truthful algorithm. We give a differentially private (and hence approximately truthful)
algorithm that achieves a logarithmic additive approximation.