Privacy-preserving query-answering systems answer queries while provably guaranteeing that sensitive information is kept secret. One very attractive notion of privacy is perfect privacy -- a secret is expressed through a query QS, and a query QV is answered only if it discloses no information about the secret query QS. However, if QS and QV are arbitrary conjunctive queries, the problem of checking whether QV discloses any information about QS is known to be p 2-complete. In this paper, we show that for large interesting subclasses of conjunctive queries enforcing perfect privacy is tractable. Instead of giving different arguments for query classes of varying complexity, we make a connection between perfect privacy and the problem of checking query containment. We then use this connection to relate the complexity of enforcing perfect privacy to the complexity of query containment.