Parallel algorithms for the maximum flow problem with minimum lot sizes

Conference paper
Part of the Operations Research Proceedings book series (ORP)


In many transportation systems, the shipment quantities are subject to minimum lot sizes in addition to regular capacity constraints. This means that either the quantity must be zero, or it must be between the two bounds. In this work, we prove that the maximum flow problem with minimum lot-size constraints on the arcs is strongly NP-hard, and we enhance the performance of a previously suggested heuristic. Profiling the serial implementation shows that most of the execution time is spent on solving a series of regular max flow problems. Therefore, we develop a parallel augmenting path algorithm that accelerates the heuristic by an average factor of 1.25.


Parallel Algorithm Start Node Main Thread Network Flow Problem Serial Implementation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, R.J., Setubal, J.C.: On the parallel implementation of Goldberg’s maximum flow algorithm. In: Proceedings of the fourth annual ACM symposium on Parallel algorithms and architectures, SPAA’92, pp. 168-177, ACM (1992)Google Scholar
  2. 2.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NPCompleteness. Freeman (1979)Google Scholar
  3. 3.
    Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM. 35, 921–940 (1988)CrossRefGoogle Scholar
  4. 4.
    Goldfarb, D., Grigoriadis, M.D.: A Computational Comparison of the Dinic and Network Simplex Methods for Maximum Flow. Annals of Operations Research. 78, 83–123 (1988)Google Scholar
  5. 5.
    Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to Parallel Computation: P-Completeness Theory. Oxford University Press (1995)Google Scholar
  6. 6.
    Haugland, D., Eleyat, M., Hetland, M.L.: The maximum flow problem with minimum lot sizes. In: Proceedings of the Second international conference on Computational logistics, ICCL’11, pp. 170-182. Springer-Verlag (2011)Google Scholar
  7. 7.
    Hong, B., He, Z.: An Asynchronous Multithreaded Algorithm for the Maximum Network Flow Problem with Nonblocking Global Relabeling Heuristic. IEEE Trans. Parallel Distrib. Syst. 22, 1025–1033 (2011)CrossRefGoogle Scholar
  8. 8.
    Jungnickel, D.: Graphs, Networks and Algorithms. Springer (2008)Google Scholar
  9. 9.
    Karzanov, A.V.: Determining the maximal flow in a network by the method of preflows. Soviet Math. Dokl. 15, 434–437 (1975)Google Scholar
  10. 10.
    Parmar, P.: Integer Programming Approaches for Equal-Split Network Flow Problems. PhD thesis, Georgia Institute of Technology (2007)Google Scholar
  11. 11.
    Shiloach, Y., Vishkin, U.: An O(n2 logn) parallel max-flow algorithm. Journal of Algorithms. 3, 128–146 (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Miriam AS, Halden, Norway & IDI, NTNUTrondheimNorway
  2. 2.Department of InformaticsUniversity of BergenBergenNorway
  3. 3.Department of Computer and Information Science (IDI)Norwegian University of Science and Technology (NTNU)TrondheimNorway

Personalised recommendations