Towards the boundary between easy and hard control problems in multicast Clos networks

In this article we study 3-stage Clos networks with multicast calls in general and 2-cast calls, in particular. We investigate various sizes of input and output switches and discuss some routing problems involved in blocking states. To express our results in a formal way we introduce a model of hypergraph edge-coloring. A new class of bipartite hypergraphs corresponding to Clos networks is studied. We identify some polynomially solvable instances as well as a number of NP-complete cases. Our results warn of possible troubles arising in the control of Clos networks even if they are composed of small-size switches in outer stages. This is in sharp contrast to classical unicast Clos networks for which all the control problems are polynomially solvable.