TY - JOUR

T1 - Private approximation of clustering and vertex cover

AU - Beimel, Amos

AU - Hallak, Renen

AU - Nissim, Kobbi

N1 - Funding Information:
Research partially supported by the Israel Science Foundation(grant No. 860/06), and by the Frankel Center for Computer Science at BGU.

PY - 2009/10/1

Y1 - 2009/10/1

N2 - 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 are allowed to leak a significant number of bits of information. For the vertex cover problem we show a tight infeasibility result: every algorithm that ρ(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 simpler and more intuitive than those in Beimel et al., and yet allow for 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.

AB - 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 are allowed to leak a significant number of bits of information. For the vertex cover problem we show a tight infeasibility result: every algorithm that ρ(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 simpler and more intuitive than those in Beimel et al., and yet allow for 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.

KW - Error-correcting reductions

KW - Hardness of unique problems

KW - Private approximation

KW - Search problems

UR - http://www.scopus.com/inward/record.url?scp=71449100498&partnerID=8YFLogxK

U2 - 10.1007/s00037-009-0259-3

DO - 10.1007/s00037-009-0259-3

M3 - Article

AN - SCOPUS:71449100498

VL - 18

SP - 435

EP - 494

JO - Computational Complexity

JF - Computational Complexity

SN - 1016-3328

IS - 3

ER -