IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i3p385-d330378.html
   My bibliography  Save this article

Mixed Graph Colorings: A Historical Review

Author

Listed:
  • Yuri N. Sotskov

    (United Institute of Informatics Problems, National Academy of Sciences of Belarus, Surganova Street 6, 220012 Minsk, Belarus)

Abstract

This paper presents a historical review and recent developments in mixed graph colorings in the light of scheduling problems with the makespan criterion. A mixed graph contains both a set of arcs and a set of edges. Two types of colorings of the vertices of the mixed graph and one coloring of the arcs and edges of the mixed graph have been considered in the literature. The unit-time scheduling problem with the makespan criterion may be interpreted as an optimal coloring of the vertices of a mixed graph, where the number of used colors is minimum. Complexity results for optimal colorings of the mixed graph are systematized. The published algorithms for finding optimal mixed graph colorings are briefly surveyed. Two new colorings of a mixed graph are introduced.

Suggested Citation

  • Yuri N. Sotskov, 2020. "Mixed Graph Colorings: A Historical Review," Mathematics, MDPI, vol. 8(3), pages 1-24, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:385-:d:330378
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/3/385/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/3/385/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Peter Brucker & M. R. Garey & D. S. Johnson, 1977. "Scheduling Equal-Length Tasks Under Treelike Precedence Constraints to Minimize Maximum Lateness," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 275-284, August.
    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. D. P. Williamson & L. A. Hall & J. A. Hoogeveen & C. A. J. Hurkens & J. K. Lenstra & S. V. Sevast'janov & D. B. Shmoys, 1997. "Short Shop Schedules," Operations Research, INFORMS, vol. 45(2), pages 288-294, April.
    4. 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.
    5. Peter Damaschke, 2019. "Parameterized Mixed Graph Coloring," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 362-374, August.
    6. M. R. Garey & D. S. Johnson & Ravi Sethi, 1976. "The Complexity of Flowshop and Jobshop Scheduling," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 117-129, May.
    7. Egon Balas, 1969. "Machine Sequencing Via Disjunctive Graphs: An Implicit Enumeration Algorithm," Operations Research, INFORMS, vol. 17(6), pages 941-957, December.
    8. Pierre Hansen & Julio Kuplinsky & Dominique Werra, 1997. "Mixed graph colorings," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 45(1), pages 145-160, February.
    9. Włodzimierz Szwarc, 1960. "Solution of the Akers-Friedman Scheduling Problem," Operations Research, INFORMS, vol. 8(6), pages 782-788, December.
    10. William W. Hardgrave & George L. Nemhauser, 1963. "A Geometric Model and a Graphical Algorithm for a Sequencing Problem," Operations Research, INFORMS, vol. 11(6), pages 889-900, December.
    11. 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.
    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. Rakhi Das & Laxminarayan Sahoo & Sovan Samanta & Vladimir Simic & Tapan Senapati, 2022. "Identifying the Shortest Path of a Semidirected Graph and Its Application," Mathematics, MDPI, vol. 10(24), pages 1-13, December.

    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. 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.
    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. Blazewicz, Jacek & Domschke, Wolfgang & Pesch, Erwin, 1996. "The job shop scheduling problem: Conventional and new solution techniques," European Journal of Operational Research, Elsevier, vol. 93(1), pages 1-33, August.
    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. Peter Damaschke, 2019. "Parameterized Mixed Graph Coloring," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 362-374, August.
    6. 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.
    7. Monaci, Marta & Agasucci, Valerio & Grani, Giorgio, 2024. "An actor-critic algorithm with policy gradients to solve the job shop scheduling problem using deep double recurrent agents," European Journal of Operational Research, Elsevier, vol. 312(3), pages 910-926.
    8. Jansen, Klaus & Mastrolilli, Monaldo & Solis-Oba, Roberto, 2005. "Approximation schemes for job shop scheduling problems with controllable processing times," European Journal of Operational Research, Elsevier, vol. 167(2), pages 297-319, December.
    9. 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.
    10. 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.
    11. Yong Chen & Yinhui Cai & Longcheng Liu & Guangting Chen & Randy Goebel & Guohui Lin & Bing Su & An Zhang, 2022. "Path cover with minimum nontrivial paths and its application in two-machine flow-shop scheduling with a conflict graph," Journal of Combinatorial Optimization, Springer, vol. 43(3), pages 571-588, April.
    12. Olivier Ploton & Vincent T’kindt, 2023. "Moderate worst-case complexity bounds for the permutation flowshop scheduling problem using Inclusion–Exclusion," Journal of Scheduling, Springer, vol. 26(2), pages 137-145, April.
    13. Souren, Rainer & Gerlach, Kurt, 2007. "Graphische Verfahren zur Maschinenbelegungsplanung: Lösungsansätze für Probleme mit zwei Aufträgen und mehrdimensionale Erweiterungen," Ilmenauer Schriften zur Betriebswirtschaftslehre, Technische Universität Ilmenau, Institut für Betriebswirtschaftslehre, volume 1, number 12007, September.
    14. Jianming Dong & Yong Chen & An Zhang & Qifan Yang, 2013. "A new three-machine shop scheduling: complexity and approximation algorithm," Journal of Combinatorial Optimization, Springer, vol. 26(4), pages 799-810, November.
    15. Xiong, Hegen & Fan, Huali & Jiang, Guozhang & Li, Gongfa, 2017. "A simulation-based study of dispatching rules in a dynamic job shop scheduling problem with batch release and extended technical precedence constraints," European Journal of Operational Research, Elsevier, vol. 257(1), pages 13-24.
    16. Yuri N. Sotskov & Omid Gholami, 2017. "Mixed graph model and algorithms for parallel-machine job-shop scheduling problems," International Journal of Production Research, Taylor & Francis Journals, vol. 55(6), pages 1549-1564, March.
    17. 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.
    18. Chong Peng & Guanglin Wu & T Warren Liao & Hedong Wang, 2019. "Research on multi-agent genetic algorithm based on tabu search for the job shop scheduling problem," PLOS ONE, Public Library of Science, vol. 14(9), pages 1-19, September.
    19. 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.
    20. D Bai & L Tang, 2010. "New heuristics for flow shop problem to minimize makespan," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(6), pages 1032-1040, June.

    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:gam:jmathe:v:8:y:2020:i:3:p:385-:d:330378. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.