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This paper aims to maximize the energy efficiency of a downlink cloud radio access network (C-RAN). Here, data is transferred from a baseband unit in the core network to several remote radio heads via a set of edge routers over capacity-limited fronthaul links. The remote radio heads then send the received signals to their users via radio access links. Both data sharing and compression-based strategies are considered for fronthaul data transfer. New mixed-integer nonlinear problems are formulated, in which the ratio of network throughput and total power consumption is maximized. These challenging problem formulations include practical constraints on routing, predefined minimum data rates, fronthaul capacity, and maximum remote radio head transmit power. By employing the successive convex quadratic programming, iterative algorithms are proposed with guaranteed convergence to the Fritz John solutions of the formulated problems. Significantly, each iteration of the proposed algorithms solves only one simple convex program. Numerical examples with practical parameters confirm that the proposed joint optimization designs markedly improve the C-RAN's energy efficiency compared to benchmark schemes. They also show that the fronthaul data-sharing strategy outperforms its compression-based counterpart in terms of energy efficiency, in both single-hop and multi-hop network scenarios.