Time-varying universal maximum flow problems

We consider a time-varying network without parallel arcs and loops, where a flow must take a certain time to traverse an arc. The transit time on an arc and the capacity of an arc are all time-varying parameters, which depend on the departure time to traverse the arc. To depart at the best time, a flow can wait at the beginning vertex of an arc, which is however limited by a timevarying vertex capacity. All those parameters, namely, the transit time, the arc capacity, and the vertex capacity, are discrete functions of time t. The problem is to find an optimal solution to send the maximum flow from the source vertex to the sink vertex, within a given time T. Moreover, we address the so-called universal maximum flow problem, which is to find a solution that remains optimal when the time limit T is truncated to any t ≤ T. We consider three variants of the problem, with waiting at a vertex being arbitrarily allowed, strictly prohibited, and bounded, respectively. Relevant algorithms are proposed, which can find optimal solutions in pseudopolynomial time.