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Relay selection is crucial in improving the performance of wireless cooperative networks. Most previous works for relay selection use the current observed channel conditions to make the relay-selection decision for the subsequent frame. However, this memoryless channel assumption is often not realistic given the time-varying nature of some mobile environments. In this paper, we consider finite-state Markov channels in the relay-selection problem. Moreover, we also incorporate adaptive modulation and coding, as well as residual relay energy in the relay-selection process. The objectives of the proposed scheme are to increase spectral efficiency, mitigate error propagation, and maximize the network lifetime. The formulation of the proposed relay-selection scheme is based on recent advances in stochastic control algorithms. The obtained relay-selection policy has an indexability property that dramatically reduces the computation and implementation complexity. In addition, there is no need for a centralized control point in the network, and relays can freely join and leave from the set of potential relays. Simulation results are presented to show the effectiveness of the proposed scheme.