Differentially Private Combinatorial Optimization

TL;DR

This paper explores the application of differential privacy to combinatorial optimization problems, demonstrating that many such problems can be approximated privately even when cryptographic privacy is impossible.

Contribution

It systematically studies differentially private algorithms for various combinatorial problems, establishing their feasibility and effectiveness.

Findings

Many combinatorial problems admit good private approximation algorithms.

Differential privacy can be achieved even when cryptographic privacy is impossible.

The study covers problems like k-median, vertex and set cover, min-cut, facility location, Steiner tree, and CPP.

Abstract

Consider 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 study the problems of vertex and set cover, min-cut, facility location, Steiner tree, and the recently introduced submodular maximization problem, "Combinatorial Public Projects" (CPP).