Minimum Cost Dynamic Flows: The Series-Parallel Case

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 υ and T, a feasible dynamic flow from s to t of value υ, obeying the time bound T, and having minimum total cost. MCDFP contains as subproblems the minimum cost maximum dynamic flow problem (fix υ to the maximum amount of flow that can be sent from s to t within time T), and the minimum cost quickest flow problem (fix T to the minimum time needed for sending υ 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. It is shown that the greedy algorithm solves MCDFP restricted to graphs in G in polynomial time.