Computational complexity of multicommodity flow problems with uniform assignment of flow

This paper treats multicommodity flow problems with a flow assignment constraint. General multicommodity flow problems require both the path set in which the commodities are assigned and the volume to which the commodities are assigned for each path. However, in this paper, we treat only problems wherein paths are required. The flow is assigned to the paths uniformly. This problem appears in state-and-time-dependent routing (STR), which is a kind of dynamic routing scheme in telecommunication networks. We show that this problem is NP-complete and remains NP-complete if the lengths (number of edges) of paths are restricted to less than or equal to two but can be solved in polynomial time if the path lengths are restricted to be less than or equal to two and the number of commodities is fixed.