The architecture and design optimization issues of the Wireless Internet-access Mesh NETwork (WIMNET) has been studied to provide a scalable wireless Internet access network. WIMNET may utilize two types of APs as wireless mesh routers to achieve the scalability with sufficient bandwidth while reducing costs. One is an expensive, programmable smart AP (SAP) that can use plural channels for wireless communications and has various functions for the Internet access. Another is an inexpensive, non-programmable conventional AP (CAP) that can use only one channel. To enhance the performance of WIMNET with a small number of costly SAPs, we have previously proposed the SAP selection algorithm of selecting a fixed number of SAPs from a given set of allocated APs. The simulation result observed the existence of a minimal SAP set providing the maximal throughput. In this paper, we propose the extension of this algorithm to finding the minimal SAP set with the maximal throughput. We verify the effectiveness of our proposal through simulations in four instances using the WIMNET simulator.