In this paper, we consider the coordinated multipoint (CoMP) transmission design for the downlink cloud radio access network (Cloud-RAN). Our design aims to optimize the set of remote radio heads (RRHs) serving each user and the precoding and transmission power to minimize the total transmission power while maintaining the fronthaul capacity and users' quality-of-service (QoS) constraints. The fronthaul capacity constraint involves a nonconvex and discontinuous function that renders the optimal exhaustive search method unaffordable for large networks. To address this challenge, we propose two low-complexity algorithms. The first pricing-based algorithm solves the underlying problem through iteratively tackling a related pricing problem while appropriately updating the pricing parameter. In the second iterative linear-relaxed algorithm, we directly address the fronthaul constraint function by iteratively approximating it with a suitable linear form using a conjugate function and solving the corresponding convex problem. For performance evaluation, we also compare our proposed algorithms with two existing algorithms in the literature. Finally, extensive numerical results are presented, which illustrate the convergence of our proposed algorithms and confirm that our algorithms significantly outperform the state-of-the-art existing algorithms.