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Scheduling unit processing time arc shutdown jobs to maximize network flow over time: complexity results

We study the problem of scheduling maintenance on arcs of a capacitated network so as to maximize the total flow from a source node to a sink node over a set of time periods. Maintenance on an arc shuts down the arc for the duration of the period in which its maintenance is scheduled, making its capacity zero for that period. A set of arcs is designated to have maintenance during the planning period, which will require each to be shut down for exactly one time period. In general this problem is known to be NP-hard. Here we identify a number of characteristics that are relevant for the complexity of instance classes. In particular, we discuss instances with restrictions on the set of arcs that have maintenance to be scheduled; series parallel networks; capacities that are balanced, in the sense that the total capacity of arcs entering a (non-terminal) node equals the total capacity of arcs leaving the node; and identical capacities on all arcs.