The single allocation problem in the interacting three-hub network

We consider the single allocation problem in the interacting three-hub network with fixed hub locations. In the single allocation hub network, the hubs are fully interconnected and each nonhub node has to be connected to exactly one of the hubs. The flows between each pair of nodes are sent using the hubs as intermediate switching points. The problem is to find an optimal allocation of nonhub nodes to the hubs which minimizes the total flow cost. We show that the single allocation problem is NP-hard as soon as the number of hubs is three, although the problem in a two-hub system has polynomial time algorithms. This paper provides a mixed integer formulation of the problem and considers the polyhedral properties of it. The formulation can also be used for the single allocation problem with fixed costs for opening links, the three-terminal cut problem, and the three-processor distribution problem. Computational experiences are reported for data given in the literature and randomly generated problems.