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A significant research effort has been devoted in recent years to the design of simple and efficient scheduling policies for input queued (IQ) and combined input output queued (CIOQ) packet switches. As a result, a number of switch control algorithms have been proposed. Among these, scheduling policies based on maximum weight matching (MWM) were identified as optimal, in the sense that they were proved to achieve 100% throughput under any admissible arrival process satisfying the strong law of large number. On the contrary, it has been recently shown that the usual MWM policies fail to guarantee 100% throughput in networks of interconnected IQ/CIOQ switches. Hence, new policies suited for networks of interconnected switches were proposed and proved to achieve 100% throughput. All of these new policies require coordination and cooperation among different switches. In this paper we address the open problem of the existence of local scheduling policies that guarantee 100% throughput in a network of IQ/CIOQ switches, providing a positive answer to such question. The only assumptions on the input traffic are that it satisfies the strong law of large numbers and that it does not oversubscribe any link in the network.