Private approximation of search problems

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.