## Abstract

Consider the following problem: given a metric space, some of whose points are "clients," select a set of at most k facility locations to minimize the average distance from the clients to their nearest facility. 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. This raises the following quandary: what if the locations of the clients are 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. [28] 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.

Original language | English |
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Title of host publication | Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms |

Publisher | Association for Computing Machinery (ACM) |

Pages | 1106-1125 |

Number of pages | 20 |

ISBN (Print) | 9780898717013 |

DOIs | |

State | Published - 2010 |

Event | 21st Annual ACM-SIAM Symposium on Discrete Algorithms - Austin, TX, United States Duration: 17 Jan 2010 → 19 Jan 2010 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Conference

Conference | 21st Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country/Territory | United States |

City | Austin, TX |

Period | 17/01/10 → 19/01/10 |