The Access Control Problem on Capacitated FIFO Networks with Unique O-D Paths is Hard

This paper is concerned with the performance of multi-commodity capacitated networks with continuous flows in a deterministic but time-dependent environment. For a given time-dependent origin-destination (O-D) table, it asks if it is easy to find a way of regulating the input flows into the network so as to avoid queues from growing in it. It is shown that even if the network structure is very simple (unique O-D paths) finding a feasible regulation scheme is a `hard' problem. More specifically, it is shown that even if all input functions are smooth, there are instances of the problem with a finite but possibly very large number of solutions. Furthermore, finding whether a particular instance of the problem has one feasible solution is an NP-hard problem because it is related to the Directed Hamiltonian Path problem of graph theory by a polynomial transformation. It is also shown that the discrete-time version of the problem is NP-complete.