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This paper is concerned with the performance of multi-commodity capacitated networks with
continuous flows in a determinsistic 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 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.
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