Multicommodity flows over time: Efficient algorithms and complexity
Theoretical Computer Science
An FPTAS for flows over time with aggregate arc capacities
WAOA'10 Proceedings of the 8th international conference on Approximation and online algorithms
Generalized maximum flows over time
WAOA'11 Proceedings of the 9th international conference on Approximation and Online Algorithms
Maximum multicommodity flows over time without intermediate storage
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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A dynamic network consists of a directed graph with capacities, costs, and integral transit times on the arcs. In the minimum-cost dynamic flow problem (MCDFP), the goal is to compute, for a given dynamic network with source s, sink t, and two integers v and T, a feasible dynamic flow from s to t of value v, obeying the time bound T, and having minimum total cost. MCDFP contains as subproblems the minimum-cost maximum dynamic flow problem, where v is fixed to the maximum amount of flow that can be sent from s to t within time T and the minimum-cost quickest flow problem, where is T is fixed to the minimum time needed for sending v units of flow from s to t. We first prove that both subproblems are NP-hard even on two-terminal series-parallel graphs with unit capacities. As main result, we formulate a greedy algorithm for MCDFP and provide a full characterization via forbidden subgraphs of the class 𝒢 of graphs, for which this greedy algorithm always yields an optimum solution (for arbitrary choices of problem parameters). 𝒢 is a subclass of the class of two-terminal series-parallel graphs. We show that the greedy algorithm solves MCDFP restricted to graphs in 𝒢 in polynomial time. © 2004 Wiley Periodicals, Inc.