Thanks to its convenience and cost-savings feature, cloud computing ushers a new era. Yet its security and privacy issues must not be neglected. Private set intersection (PSI) is useful and important in many cloud computing applications, such as document similarity, genetic paternity and data mining. The cloud server performs intersection operations on two outsourced encrypted datasets of data owners. In the existing protocols, however, data owners cannot decide whether to use all or part of their encrypted data to compute the intersection, nor can they specify whom to compare with. In this paper, we introduce an enhanced notion of outsourced PSI, called authorized PSI (APSI), which supports flexible authorization and cross-type authorized comparison of datasets. To demonstrate this notion, we propose a concrete APSI protocol, and prove it to be secure in the random oracle model based on simple number-theoretic assumptions. Experimental results show that our APSI protocol has performance comparable with existing related outsourced PSI protocols.