IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v229y2015i1p397-40810.1007-s10479-015-1807-7.html
   My bibliography  Save this article

Solving the parametric bipartite maximum flow problem in unbalanced and closure bipartite graphs

Author

Listed:
  • Mehdi Ghiyasvand

Abstract

Given a directed bipartite graph $$G=(V,E)$$ G = ( V , E ) with a node set $$V=\{s\} \cup V_1 \cup V_2 \cup \{t\}$$ V = { s } ∪ V 1 ∪ V 2 ∪ { t } , and an arc set $$E=E_1\cup E_2\cup E_3$$ E = E 1 ∪ E 2 ∪ E 3 , where $$E_1=\{s\} \times V_1$$ E 1 = { s } × V 1 , $$E_2= V_1 \times V_2$$ E 2 = V 1 × V 2 , and $$E_3=V_2 \times \{t\}$$ E 3 = V 2 × { t } . Chen ( 1995 ) presented an $$O(|V| |E| \log (\frac{|V|^2}{|E|}))$$ O ( | V | | E | log ( | V | 2 | E | ) ) time algorithm to solve the parametric bipartite maximum flow problem. In this paper, we assume all arcs in $$E_2$$ E 2 have infinite capacity [such a graph is called closure graph Hochbaum ( 1998 )], and present a new approach to solve the problem, which runs in $$O(|V_1| |E| \log (\frac{|V_1|^2}{|E|}+2))$$ O ( | V 1 | | E | log ( | V 1 | 2 | E | + 2 ) ) time using Gallo et al’s parametric maximum flow algorithm, see Gallo et al. ( 1989 ). In unbalanced bipartite graphs, we have $$|V_1| >> |V_2|$$ | V 1 | > > | V 2 | , so, our algorithm improves Chens’s algorithm in unbalanced and closure bipartite graphs. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Mehdi Ghiyasvand, 2015. "Solving the parametric bipartite maximum flow problem in unbalanced and closure bipartite graphs," Annals of Operations Research, Springer, vol. 229(1), pages 397-408, June.
    • Handle: RePEc:spr:annopr:v:229:y:2015:i:1:p:397-408:10.1007/s10479-015-1807-7
    DOI: 10.1007/s10479-015-1807-7
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10479-015-1807-7
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10479-015-1807-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Sedeño-Noda, A. & de Pablo, D. Alcaide López & González-Martín, C., 2009. "A network flow-based method to solve performance cost and makespan open-shop scheduling problems with time-windows," European Journal of Operational Research, Elsevier, vol. 196(1), pages 140-154, July.
    2. Bruce Faaland & Tom Schmitt, 1987. "Scheduling Tasks with Due Dates in a Fabrication/Assembly Process," Operations Research, INFORMS, vol. 35(3), pages 378-388, June.
    3. Robert McNaughton, 1959. "Scheduling with Deadlines and Loss Functions," Management Science, INFORMS, vol. 6(1), pages 1-12, October.
    4. Sedeno-Noda, A. & Alcaide, D. & Gonzalez-Martin, C., 2006. "Network flow approaches to pre-emptive open-shop scheduling problems with time-windows," European Journal of Operational Research, Elsevier, vol. 174(3), pages 1501-1518, November.
    5. Chen, Y. L., 1995. "A parametric maximum flow algorithm for bipartite graphs with applications," European Journal of Operational Research, Elsevier, vol. 80(1), pages 226-235, January.
    6. W. A. Horn, 1974. "Some simple scheduling algorithms," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 21(1), pages 177-185, March.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Akiyoshi Shioura & Natalia V. Shakhlevich & Vitaly A. Strusevich, 2017. "Machine Speed Scaling by Adapting Methods for Convex Optimization with Submodular Constraints," INFORMS Journal on Computing, INFORMS, vol. 29(4), pages 724-736, November.
    2. Akiyoshi Shioura & Natalia V. Shakhlevich & Vitaly A. Strusevich, 2020. "Scheduling problems with controllable processing times and a common deadline to minimize maximum compression cost," Journal of Global Optimization, Springer, vol. 76(3), pages 471-490, March.
    3. Sedeño-Noda, A. & de Pablo, D. Alcaide López & González-Martín, C., 2009. "A network flow-based method to solve performance cost and makespan open-shop scheduling problems with time-windows," European Journal of Operational Research, Elsevier, vol. 196(1), pages 140-154, July.
    4. Johnny C. Ho & Yih‐Long Chang, 1991. "Heuristics for minimizing mean tardiness for m parallel machines," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(3), pages 367-381, June.
    5. Joseph Y.‐T. Leung & Michael Pinedo, 2004. "A note on scheduling parallel machines subject to breakdown and repair," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(1), pages 60-71, February.
    6. T.C.E. Cheng & Svetlana A. Kravchenko & Bertrand M.T. Lin, 2017. "Preemptive parallel‐machine scheduling with a common server to minimize makespan," Naval Research Logistics (NRL), John Wiley & Sons, vol. 64(5), pages 388-398, August.
    7. Liu Guiqing & Li Kai & Cheng Bayi, 2015. "Preemptive Scheduling with Controllable Processing Times on Parallel Machines," Journal of Systems Science and Information, De Gruyter, vol. 3(1), pages 68-76, February.
    8. Hoogeveen, J. A. & Lenstra, J. K. & Veltman, B., 1996. "Preemptive scheduling in a two-stage multiprocessor flow shop is NP-hard," European Journal of Operational Research, Elsevier, vol. 89(1), pages 172-175, February.
    9. J. Steve Davis & John J. Kanet, 1993. "Single‐machine scheduling with early and tardy completion costs," Naval Research Logistics (NRL), John Wiley & Sons, vol. 40(1), pages 85-101, February.
    10. Yung-Chia Chang & Kuei-Hu Chang & Ching-Ping Zheng, 2022. "Application of a Non-Dominated Sorting Genetic Algorithm to Solve a Bi-Objective Scheduling Problem Regarding Printed Circuit Boards," Mathematics, MDPI, vol. 10(13), pages 1-21, July.
    11. Fowler, John W. & Mönch, Lars, 2022. "A survey of scheduling with parallel batch (p-batch) processing," European Journal of Operational Research, Elsevier, vol. 298(1), pages 1-24.
    12. Scholl, Armin & Becker, Christian, 2006. "State-of-the-art exact and heuristic solution procedures for simple assembly line balancing," European Journal of Operational Research, Elsevier, vol. 168(3), pages 666-693, February.
    13. C N Potts & V A Strusevich, 2009. "Fifty years of scheduling: a survey of milestones," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 60(1), pages 41-68, May.
    14. Lin, Hung-Tso & Lee, Hong-Tau & Pan, Wen-Jung, 2008. "Heuristics for scheduling in a no-wait open shop with movable dedicated machines," International Journal of Production Economics, Elsevier, vol. 111(2), pages 368-377, February.
    15. Leah Epstein, 2023. "Parallel solutions for preemptive makespan scheduling on two identical machines," Journal of Scheduling, Springer, vol. 26(1), pages 61-76, February.
    16. Djellab, Housni & Djellab, Khaled, 2002. "Preemptive Hybrid Flowshop Scheduling problem of interval orders," European Journal of Operational Research, Elsevier, vol. 137(1), pages 37-49, February.
    17. Han, Bin & Zhang, Wenjun & Lu, Xiwen & Lin, Yingzi, 2015. "On-line supply chain scheduling for single-machine and parallel-machine configurations with a single customer: Minimizing the makespan and delivery cost," European Journal of Operational Research, Elsevier, vol. 244(3), pages 704-714.
    18. Sedeno-Noda, A. & Alcaide, D. & Gonzalez-Martin, C., 2006. "Network flow approaches to pre-emptive open-shop scheduling problems with time-windows," European Journal of Operational Research, Elsevier, vol. 174(3), pages 1501-1518, November.
    19. Huo, Yumei & Zhao, Hairong, 2015. "Total completion time minimization on multiple machines subject to machine availability and makespan constraints," European Journal of Operational Research, Elsevier, vol. 243(2), pages 547-554.
    20. Alexander Grigoriev & Martijn Holthuijsen & Joris van de Klundert, 2005. "Basic scheduling problems with raw material constraints," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(6), pages 527-535, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:annopr:v:229:y:2015:i:1:p:397-408:10.1007/s10479-015-1807-7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.