–An FPGA switch box is said to be hyper-universal if it is routable for all possible surrounding multi-pin net topologies satisfying the routing resource constraints. It is desirable to design hyper-universal switch boxes with the minimum number of switches. A previous work, Universal Switch Module, considered such a design problem concerning -pin net routings around a single FPGA switch box. However, as most nets are multi-pin nets in practice, it is imperative to study the problem that involves multi-pin nets. In this paper, we provide a new view of global routings and formulate the most general -sided switch box design problem into an optimum -partite graph design problem. Applying a powerful decomposition theorem of global routings, we prove that, for a fixed , the number of switches in an optimum -sided switch box with terminals on each side is , by constructing some hyper-universal switch boxes with switches. Furthermore, we obtain optimum, hyper-universal sided and -sided swi...