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 a 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 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 a 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 yield an optimum solution (for arbitrary choices of problem parameters). 𝒢 is a subclass of two-terminal series-parallel graphs. We show that the greedy algorithm solves MCDFP restricted to graphs in 𝒢 in polynomial time.
