TY - JOUR

T1 - Private approximation of search problems

AU - Beimel, Amos

AU - Carmi, Paz

AU - Nissim, Kobbi

AU - Weinreb, Enav

N1 - Funding Information:
Supported by awards from the NIH (AI28571 to B.M.D.), from the National Cancer Institute, DHHS, under Contract No. N01-C074101 with A.B.L. (to A.W.), and from the MRC-AIDS Directed Program (to J.K.). We are most grateful to our many colleagues, past and present, in our own and in other laboratories and too numerous to acknowledge individually, who have contributed substantially to the work described. We thank Dr. Jacek Lubkowski (NCI-Frederick) for preparing Fig. 1.

PY - 2008/12/29

Y1 - 2008/12/29

N2 - Many approximation algorithms have been presented in the last decades for NP-hard search problems. The focus of this paper is on cryptographic applications, where it is desirable to design algorithms which do not leak unnecessary information. Specifically, we are interested in private approximation algorithms - efficient approximation algorithms whose output does not leak information not implied by the optimal solutions to the search problems. Privacy requirements add constraints on the approximation algorithms; in particular, known approximation algorithms usually leak a lot of information. For functions, Feigenbaum et al. [ACM Trans. Algorithms, 2 (2006), pp. 435-472] presented a natural requirement that a private algorithm should not leak information not implied by the original function. Generalizing this requirement to relations is not straightforward as an input may have many different outputs. We present a new definition that captures a minimal privacy requirement from such algorithms; applied to an input instance, it should not leak any information that is not implied by its collection of exact solutions. We argue that our privacy requirement is natural and quite minimal. We show that, even under this minimal definition of privacy, for well-studied problems such as vertex cover and max exact 3SAT, private approximation algorithms are unlikely to exist even for poor approximation ratios. Similarly to Halevi et al. [in Proceedings of the 33rd ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 550-559], we define a relaxed notion of approximation algorithms that leak (a little) information, and demonstrate the applicability of this notion by showing near optimal approximation algorithms for max exact 3SAT that leak a little information.

AB - Many approximation algorithms have been presented in the last decades for NP-hard search problems. The focus of this paper is on cryptographic applications, where it is desirable to design algorithms which do not leak unnecessary information. Specifically, we are interested in private approximation algorithms - efficient approximation algorithms whose output does not leak information not implied by the optimal solutions to the search problems. Privacy requirements add constraints on the approximation algorithms; in particular, known approximation algorithms usually leak a lot of information. For functions, Feigenbaum et al. [ACM Trans. Algorithms, 2 (2006), pp. 435-472] presented a natural requirement that a private algorithm should not leak information not implied by the original function. Generalizing this requirement to relations is not straightforward as an input may have many different outputs. We present a new definition that captures a minimal privacy requirement from such algorithms; applied to an input instance, it should not leak any information that is not implied by its collection of exact solutions. We argue that our privacy requirement is natural and quite minimal. We show that, even under this minimal definition of privacy, for well-studied problems such as vertex cover and max exact 3SAT, private approximation algorithms are unlikely to exist even for poor approximation ratios. Similarly to Halevi et al. [in Proceedings of the 33rd ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 550-559], we define a relaxed notion of approximation algorithms that leak (a little) information, and demonstrate the applicability of this notion by showing near optimal approximation algorithms for max exact 3SAT that leak a little information.

KW - Private approximation

KW - Secure computation

KW - Solution-list algorithms

KW - Vertex cover

UR - http://www.scopus.com/inward/record.url?scp=57849121671&partnerID=8YFLogxK

U2 - 10.1137/060671899

DO - 10.1137/060671899

M3 - Article

AN - SCOPUS:57849121671

SN - 0097-5397

VL - 38

SP - 1728

EP - 1760

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 5

ER -