IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v3y1999i4d10.1023_a1009827520712.html
   My bibliography  Save this article

Transversal Graphs for Partially Ordered Sets: Sequencing, Merging and Scheduling Problems

Author

Listed:
  • Martin Middendorf

    (University of Karlsruhe)

  • Vadim G. Timkovsky

    (Star Data Systems INc.
    McMaster University)

Abstract

This paper introduces an approach to solving combinatorial optimization problems on partially ordered sets by the reduction to searching source-sink paths in the related transversal graphs. Different techniques are demonstrated in application to finding consistent supersequences, merging partially ordered sets, and machine scheduling with precedence constraints. Extending the approach to labeled partially ordered sets we also propose a solution for the smallest superplan problem and show its equivalence to the well studied coarsest regular refinement problem. For partially ordered sets of a fixed width the number of vertices in their transversal graphs is polynomial, so the reduction allows us easily to establish that many related problems are solvable in polynomial or pseudopolynomial time. For example, we establish that the longest consistent supersequence problem with a fixed number of given strings can be solved in polynomial time, and that the precedence-constrained release-date maximum- or total-cost preemptive or nonpreemptive job-shop scheduling problem with a fixed number of jobs can be solved in pseudopolynomial time. We also show that transversal graphs can be used to generalize and strengthen similar results obtained earlier by dynamic programming.

Suggested Citation

  • Martin Middendorf & Vadim G. Timkovsky, 1999. "Transversal Graphs for Partially Ordered Sets: Sequencing, Merging and Scheduling Problems," Journal of Combinatorial Optimization, Springer, vol. 3(4), pages 417-435, December.
    • Handle: RePEc:spr:jcomop:v:3:y:1999:i:4:d:10.1023_a:1009827520712
    DOI: 10.1023/A:1009827520712
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1009827520712
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1009827520712?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. Peter Brucker & Svetlana A. Kravchenko & Yuri N. Sotskov, 1999. "Preemptive job-shop scheduling problems with a fixed number of jobs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(1), pages 41-76, March.
    2. Brucker, Peter & Kramer, Andreas, 1996. "Polynomial algorithms for resource-constrained and multiprocessor task scheduling problems," European Journal of Operational Research, Elsevier, vol. 90(2), pages 214-226, April.
    3. Sotskov, Yu. N., 1991. "The complexity of shop-scheduling problems with two or three jobs," European Journal of Operational Research, Elsevier, vol. 53(3), pages 326-336, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Timkovsky, Vadim G., 2003. "Identical parallel machines vs. unit-time shops and preemptions vs. chains in scheduling complexity," European Journal of Operational Research, Elsevier, vol. 149(2), pages 355-376, September.
    2. D. Prot & O. Bellenguez-Morineau, 2018. "A survey on how the structure of precedence constraints may change the complexity class of scheduling problems," Journal of Scheduling, Springer, vol. 21(1), pages 3-16, February.

    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. Yuri N. Sotskov, 2020. "Mixed Graph Colorings: A Historical Review," Mathematics, MDPI, vol. 8(3), pages 1-24, March.
    2. Peter Brucker & Yu Sotskov & Frank Werner, 2007. "Complexity of shop-scheduling problems with fixed number of jobs: a survey," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(3), pages 461-481, June.
    3. Jacomine Grobler & Andries Engelbrecht & Schalk Kok & Sarma Yadavalli, 2010. "Metaheuristics for the multi-objective FJSP with sequence-dependent set-up times, auxiliary resources and machine down time," Annals of Operations Research, Springer, vol. 180(1), pages 165-196, November.
    4. Agnetis, Alessandro & Kellerer, Hans & Nicosia, Gaia & Pacifici, Andrea, 2012. "Parallel dedicated machines scheduling with chain precedence constraints," European Journal of Operational Research, Elsevier, vol. 221(2), pages 296-305.
    5. Geurtsen, M. & Didden, Jeroen B.H.C. & Adan, J. & Atan, Z. & Adan, I., 2023. "Production, maintenance and resource scheduling: A review," European Journal of Operational Research, Elsevier, vol. 305(2), pages 501-529.
    6. Jain, A. S. & Meeran, S., 1999. "Deterministic job-shop scheduling: Past, present and future," European Journal of Operational Research, Elsevier, vol. 113(2), pages 390-434, March.
    7. Peter Damaschke, 2019. "Parameterized Mixed Graph Coloring," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 362-374, August.
    8. Xiaoqiang Cai & Chung‐Yee Lee & Chung‐Lun Li, 1998. "Minimizing total completion time in two‐processor task systems with prespecified processor allocations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 45(2), pages 231-242, March.
    9. Jianer Chen & Chung‐Yee Lee, 1999. "General multiprocessor task scheduling," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(1), pages 57-74, February.
    10. Y N Sotskov & A Allahverdi & T-C Lai, 2004. "Flowshop scheduling problem to minimize total completion time with random and bounded processing times," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(3), pages 277-286, March.
    11. Abdeljaouad, Mohamed Amine & Bahroun, Zied & Omrane, Anissa & Fondrevelle, Julien, 2015. "Job-shop production scheduling with reverse flows," European Journal of Operational Research, Elsevier, vol. 244(1), pages 117-128.
    12. Lei Lei & Michael Pinedo & Lian Qi & Shengbin Wang & Jian Yang, 2015. "Personnel scheduling and supplies provisioning in emergency relief operations," Annals of Operations Research, Springer, vol. 235(1), pages 487-515, December.
    13. Timkovsky, Vadim G., 2003. "Identical parallel machines vs. unit-time shops and preemptions vs. chains in scheduling complexity," European Journal of Operational Research, Elsevier, vol. 149(2), pages 355-376, September.
    14. Edis, Emrah B. & Oguz, Ceyda & Ozkarahan, Irem, 2013. "Parallel machine scheduling with additional resources: Notation, classification, models and solution methods," European Journal of Operational Research, Elsevier, vol. 230(3), pages 449-463.
    15. Boros, Endre & Gurvich, Vladimir, 2000. "Stable effectivity functions and perfect graphs," Mathematical Social Sciences, Elsevier, vol. 39(2), pages 175-194, March.
    16. Peleg, Bezalel & Peters, Hans & Storcken, Ton, 2002. "Nash consistent representation of constitutions: a reaction to the Gibbard paradox," Mathematical Social Sciences, Elsevier, vol. 43(2), pages 267-287, March.
    17. Shakhlevich, Natalia V. & Sotskov, Yuri N. & Werner, Frank, 2000. "Complexity of mixed shop scheduling problems: A survey," European Journal of Operational Research, Elsevier, vol. 120(2), pages 343-351, January.
    18. Oguz, Ceyda & Zinder, Yakov & Ha Do, Van & Janiak, Adam & Lichtenstein, Maciej, 2004. "Hybrid flow-shop scheduling problems with multiprocessor task systems," European Journal of Operational Research, Elsevier, vol. 152(1), pages 115-131, January.
    19. Mati, Yazid & Xie, Xiaolan, 2004. "The complexity of two-job shop problems with multi-purpose unrelated machines," European Journal of Operational Research, Elsevier, vol. 152(1), pages 159-169, January.
    20. F Sivrikaya şerifoğlu & G Ulusoy, 2004. "Multiprocessor task scheduling in multistage hybrid flow-shops: a genetic algorithm approach," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(5), pages 504-512, May.

    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:jcomop:v:3:y:1999:i:4:d:10.1023_a:1009827520712. 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.