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This paper considers downlink multi-user transmission facilitated by a reconfigurable intelligent surface (RIS). First, focusing on the multi-group multicast beamforming scenario, we develop a fast and scalable algorithm for the joint base station (BS) and RIS beamforming optimization to minimize the transmit power subject to the user quality-of-service (QoS) constraints. By exploring the structure of this QoS problem, we show that the joint beamforming optimization can be naturally decomposed into a BS multicast beamforming QoS problem and an RIS passive multicast beamforming max-min-fair (MMF) problem. We propose an alternating multicast beamforming (AMBF) algorithm to solve the two subproblems alternatingly. For the BS QoS subproblem, we utilize the optimal multicast beamforming structure to obtain the BS beamformers efficiently. Furthermore, we reformulate the challenging RIS MMF subproblem and employ a first-order projected subgradient algorithm (PSA), which yields closed-form updates. The computational complexity of the AMBF algorithm grows linearly with the number of RIS elements and BS antennas. We further show that the AMBF approach is also an efficient method for the RIS-assisted downlink multi-user unicast beamforming problem, providing semi-closed-form updates. Next, we study the MMF problem for the RIS-assisted downlink beamforming design and propose a PSA-based fast algorithm to compute the BS and RIS beamforming solutions with closed-form updates per iteration, leading to a highly computationally efficient solution. Simulation results show the efficacy of our proposed algorithms in both performance and computational cost compared to other alternative methods., Comment: 13 pages, 10 figures