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.
KeywordsDynamic Network Sink Node Polynomial Time Algorithm Capacity Function Node Capacity
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