Private approximation of clustering and vertex cover

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/ρ(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.