In order to handle a high traffic demand, dense wireless local area networks (WLANs) have been deployed rapidly in the past years. However, dense WLANs cause two critical issues: wastage of energy and severe interference. To address these issues, the centralized management of dense WLANs has been emerged as a powerful paradigm for improving energy efficiency as well as avoiding severe interference. In this paper, we study the joint optimization problem of power-operation modes in access points (APs), channel selections and user-AP associations for improving energy efficiency and avoiding interference without sacrificing users' demands. To this end, we first formulate it as a mixed-integer programming using the popular Lyapunov approach, but it turns out to be computationally intractable, i.e., NP-hard. To address the issue, we propose a polynomial-time approximation algorithm and prove that it achieves a constant-factor approximation guarantee under mild assumptions. The main novelty underlying our algorithm design is based on a linear programming relaxation combining with two different greedy rounding schemes, where each achieves a constant-factor approximation in different regimes of parameters. We verify the performance of the proposed algorithm via extensive simulations and also demonstrate its practicability by implementing it at commercial APs using a Software-defined Networking framework. Results from our experiments show that it reduces the wasted energy significantly while maintaining even higher throughput.