As an inexpensive, flexible, and scalable Internet-access wireless network, we have studied the architecture, protocols, and design optimizations of the Wireless Internet-access Mesh NETwork (WIMNET) that adopts multiple access-points (APs) connected through wireless links. WIMNET can improve the dependability to failures of APs and/or their links by allocating APs redundantly in the network field. Because redundant APs may increase the operational cost and degrade the throughput due to increasing radio interferences, only the necessary APs for connectivity between the hosts and the Internet gateway should be activated in communications. In this paper, we first define the active AP selection problem of selecting the minimum number of active APs, and prove the NP-completeness of its decision version through reduction from the known minimum set cover problem. Then, we propose an active AP selection algorithm of deactivating APs one by one until no more AP can be deactivated. Due to the limited transmission capacity in WIMNET, we further present an algorithm extension for throughput maximization. We verify the effectiveness of our proposal through simulations in three instances, where the number of active APs is reduced by more than 40% and the throughput is improved by more than 10% from the original in any instance.