A unified approach to approximation algorithms for bottleneck problems
Journal of the ACM (JACM)
NP is as easy as detecting unique solutions
Theoretical Computer Science
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Completeness theorems for non-cryptographic fault-tolerant distributed computation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Private approximation of NP-hard functions
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
On the Difference Between One and Many (Preliminary Version)
Proceedings of the Fourth Colloquium on Automata, Languages and Programming
Proofs, Codes, and Polynomial-Time Reducibilities
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Practical privacy: the SuLQ framework
Proceedings of the twenty-fourth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Private approximation of search problems
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Secure multiparty computation of approximations
ACM Transactions on Algorithms (TALG)
Protocols for secure computations
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
Secure computation of the mean and related statistics
TCC'05 Proceedings of the Second international conference on Theory of Cryptography
Polylogarithmic private approximations and efficient matching
TCC'06 Proceedings of the Third conference on Theory of Cryptography
Private multiparty sampling and approximation of vector combinations
Theoretical Computer Science
Longest common subsequence as private search
Proceedings of the 8th ACM workshop on Privacy in the electronic society
How should we solve search problems privately?
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Differentially private combinatorial optimization
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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Private approximation of search problems deals with finding approximate solutions to search problems while disclosing as little information as possible. The focus of this work is on private approximation of the vertex cover problem and two well studied clustering problems - k-center and k-median. Vertex cover was considered in [Beimel, Carmi, Nissim, and Weinreb, STOC, 2006] and we improve their infeasibility results. Clustering algorithms are frequently applied to sensitive data, and hence are of interest in the contexts of secure computation and private approximation. We show that these problems do not admit private approximations, or even approximation algorithms that leak significant number of bits. For the vertex cover problem we show a tight infeasibility result: every algorithm that p(n)-approximates vertex-cover must leak ω(n/p(n)) bits (where n is the number of vertices in the graph). For the clustering problems we prove that even approximation algorithms with a poor approximation ratio must leak ω(n) bits (where n is the number of points in the instance). For these results we develop new proof techniques, which are more simple and intuitive than those in Beimel et al., and yet allow stronger infeasibility results. Our proofs rely on the hardness of the promise problem where a unique optimal solution exists [Valiant and Vazirani, Theoretical Computer Science, 1986], on the hardness of approximating witnesses for NP-hard problems ([Kumar and Sivakumar, CCC, 1999] and [Feige, Langberg, and Nissim, APPROX, 2000]), and on a simple random embedding of instances into bigger instances.