Multicommodity Flows in Simple Multistage Networks Multicommodity Flows in Simple Multistage Networks

In this paper we consider the integral multicommodity ow problem on directed graphs underlying two classes of multistage interconnection networks. In one direction, we consider 3-stage networks. Using existing results on (g; f)-factors of bipartite graphs, we show suucient and necessary conditions for the existence of a solution when the network has at most 2 secondary switches. In contrast, the problem is shown to be NP-complete if the network has 3 or more secondaries. In a second direction, we introduce a recursive class of networks that includes multistage hypercubic networks (such as the omega network, the indirect binary n-cube, and the generalized cube network) as a proper subset. Networks in the new class may have arbitrary number of stages. Moreover, each stage may contain identical switches of any arbitrary size. The notion of extra-stage networks is extended to the new class, and the problem is shown to have polynomial time solutions on r-stage networks where r = 3, or where each link has a unit capacity and r 3. The latter result implies an eecient algorithm for deciding admissible permutations on conventional extra-stage hypercubic networks. In contrast, we show that the multicommodity ow problem is NP-complete on extra-stage networks, even if r = 6, each link has an integral capacity 3, and all ow demands are equal.