Minimum cost dynamic flows: The series-parallel case

Bettina Klinz, Gerhard Woeginger

Research output: Contribution to journalArticleAcademicpeer-review

56 Citations (Scopus)
64 Downloads (Pure)


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 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.
Original languageUndefined
Pages (from-to)153-162
Number of pages9
Issue number3
Publication statusPublished - 2004


  • METIS-219742
  • two-terminal series parallel networks
  • dynamic network flow
  • Greedy algorithm
  • Minimum-cost flow
  • IR-72005

Cite this