Minimum‐cost dynamic flows: The series‐parallel case

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Abstract
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.

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