Universally maximum flow with piecewise‐constant capacities

The maximum dynamic flow problem generalizes the standard maximum flow problem by introducing time. The object is to send as much flow from source to sink in T time units as possible, where capacities are interpreted as an upper bound on the rate of flow entering an arc. A related problem is the universally maximum flow, which is to send a flow from source to sink that maximizes the amount of flow arriving at the sink by time t simultaneously for all t ≤ T. We consider a further generalization of this problem that allows arc and node capacities to change over time. In particular, given a network with arc and node capacities that are piecewise constant functions of time with at most k breakpoints, and a time bound T, we show how to compute a flow that maximizes the amount of flow reaching the sink in all time intervals (0, t] simultaneously for all 0 < t ≤ T, in O(k 2 mnlog(kn 2/m)) time. The best previous algorithm requires O(nk) maximum flow computations on a network with (m+ n)k arcs and nk nodes.

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