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This paper focuses on the relay selection problem in amplify-and-forward (AF) cooperative communication with full-duplex (FD) operation. Different relay selection schemes assuming the availability of different instantaneous information are studied. We consider optimal relay selection that maximizes the instantaneous FD channel capacity and requires global channel state information (CSI) as well as several sub-optimal relay selection policies that utilize partial CSI knowledge such as a) source-relay and relay-destination links b) loop interference c) source-relay links and loop interference. To facilitate comparison, exact outage probability expressions and asymptotic approximations of these policies that show a zero diversity order are derived. In addition, an optimal relay selection procedure that incorporates a hybrid relaying strategy, which dynamically switches between FD and half-duplex relaying according to the instantaneous CSI, is also investigated.