A polynomial time algorithm for the minimum ?ow problem in time-varying networks

Abstract Flow variations over time generalize standard network ?ows by introducing an element of time. In contrast to the classical case of static ?ows, a ?ow over time in such a networkspeci?esa?owrateenteringanarcforeachpointintime.Inthissetting,thecapacity of an arc limits the rate of ?ow into the arc at each point in time. Traditionally, ?ows over time are computed in time-expanded networks that contain one copy of the original network foreachdiscretetimestep.Whilethismethodmakesavailablethewholealgorithmictoolbox developed for static network ?ows, its drawback is the enormous size of the time-expanded network. In this paper, we extend the results about the minimum ?ow problem to network ?ows (with n nodes and m arcs) in which the time-varying lower bounds can involve both the source and the sink nodes (as in Fathabadi et al.) and also one additional node other than the source and the sink nodes. It is shown that this problem for the set{0,1,...,T}of time points can be solved by at most n minimum ?ow computations, by suitably extending the dynamic minimum ?ow algorithm and reoptimization techniques. The running time of the presented algorithm is O(n2m)