Many approximation algorithms have been presented in the last decades
for hard search problems. The focus of this paper is on cryptographic
applications, where it is desired to design algorithms which do not
leak unnecessary information. Specifically, we are interested in
private approximation algorithms -- efficient 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., ICALP 2001] presented a natural
requirement that a private algorithm should not leak information not
implied by the original function. Generalizing this requirement to earch
problems is not straight forward 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. Although our privacy requirement seems minimal, we
show that for well studied problems, as vertex cover and maximum exact
3SAT, private approximation algorithms are unlikely to exist even for
poor approximation ratios. Similar to [Halevi et al., STOC 2001], we
define a relaxed notion of approximation algorithms that leak (little)
information, and demonstrate the applicability of this notion by
showing near optimal approximation algorithms for maximum exact 3SAT
which leak little information.