IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v179y2010i1p105-13010.1007-s10479-010-0716-z.html
   My bibliography  Save this article

A supernodal formulation of vertex colouring with applications in course timetabling

Author

Listed:
  • Edmund Burke
  • Jakub Mareček
  • Andrew Parkes
  • Hana Rudová

Abstract

For many problems in scheduling and timetabling, the choice of a mathematical programming formulation is determined by the formulation of the graph colouring component. This paper briefly surveys seven known integer programming formulations of vertex colouring and introduces a new approach using “supernodes”. In the definition of George and McIntyre (SIAM J. Numer. Anal. 15(1):90–112, 1978 ), a “supernode” is a complete subgraph, within which every pair of vertices have the same neighbourhood outside of the subgraph. A polynomial-time algorithm for obtaining the best possible partition of an arbitrary graph into supernodes is given. This makes it possible to use any formulation of vertex multicolouring to encode vertex colouring. Results of empirical tests on benchmark instances in graph colouring (DIMACS) and timetabling (Udine Course Timetabling) are also provided and discussed. Copyright Springer Science+Business Media, LLC 2010

Suggested Citation

  • Edmund Burke & Jakub Mareček & Andrew Parkes & Hana Rudová, 2010. "A supernodal formulation of vertex colouring with applications in course timetabling," Annals of Operations Research, Springer, vol. 179(1), pages 105-130, September.
    • Handle: RePEc:spr:annopr:v:179:y:2010:i:1:p:105-130:10.1007/s10479-010-0716-z
    DOI: 10.1007/s10479-010-0716-z
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10479-010-0716-z
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10479-010-0716-z?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. Burke, Edmund Kieran & Petrovic, Sanja, 2002. "Recent research directions in automated timetabling," European Journal of Operational Research, Elsevier, vol. 140(2), pages 266-280, July.
    2. H.P. Williams & Hong Yan, 2001. "Representations of the all_different Predicate of Constraint Satisfaction in Integer Programming," INFORMS Journal on Computing, INFORMS, vol. 13(2), pages 96-103, May.
    3. Pablo Coll & Javier Marenco & Isabel Méndez Díaz & Paula Zabala, 2002. "Facets of the Graph Coloring Polytope," Annals of Operations Research, Springer, vol. 116(1), pages 79-90, October.
    4. Anuj Mehrotra & Michael A. Trick, 1996. "A Column Generation Approach for Graph Coloring," INFORMS Journal on Computing, INFORMS, vol. 8(4), pages 344-354, November.
    5. Karen Aardal & Stan Hoesel & Arie Koster & Carlo Mannino & Antonio Sassano, 2007. "Models and solution techniques for frequency assignment problems," Annals of Operations Research, Springer, vol. 153(1), pages 79-129, September.
    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. Andrea Schaerf, 2015. "Comments on: An overview of curriculum-based course timetabling," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 362-365, July.
    2. Casado, Silvia & Laguna, Manuel & Pacheco, Joaquín & Puche, Julio C., 2020. "Grouping products for the optimization of production processes: A case in the steel manufacturing industry," European Journal of Operational Research, Elsevier, vol. 286(1), pages 190-202.
    3. Ivo Blöchliger & Nicolas Zufferey, 2013. "Multi-coloring and job-scheduling with assignment and incompatibility costs," Annals of Operations Research, Springer, vol. 211(1), pages 83-101, December.
    4. Bagger, Niels-Christian F. & Sørensen, Matias & Stidsen, Thomas R., 2019. "Dantzig–Wolfe decomposition of the daily course pattern formulation for curriculum-based course timetabling," European Journal of Operational Research, Elsevier, vol. 272(2), pages 430-446.
    5. Johnes, Jill, 2015. "Operational Research in education," European Journal of Operational Research, Elsevier, vol. 243(3), pages 683-696.
    6. Niels-Christian Fink Bagger & Guy Desaulniers & Jacques Desrosiers, 2019. "Daily course pattern formulation and valid inequalities for the curriculum-based course timetabling problem," Journal of Scheduling, Springer, vol. 22(2), pages 155-172, April.
    7. Edmund Burke & Jakub Mareček & Andrew Parkes & Hana Rudová, 2012. "A branch-and-cut procedure for the Udine Course Timetabling problem," Annals of Operations Research, Springer, vol. 194(1), pages 71-87, April.
    8. Dennis S. Holm & Rasmus Ø. Mikkelsen & Matias Sørensen & Thomas J. R. Stidsen, 2022. "A graph-based MIP formulation of the International Timetabling Competition 2019," Journal of Scheduling, Springer, vol. 25(4), pages 405-428, August.
    9. Mutsunori Banbara & Katsumi Inoue & Benjamin Kaufmann & Tenda Okimoto & Torsten Schaub & Takehide Soh & Naoyuki Tamura & Philipp Wanko, 2019. "$${\varvec{teaspoon}}$$ teaspoon : solving the curriculum-based course timetabling problems with answer set programming," Annals of Operations Research, Springer, vol. 275(1), pages 3-37, April.
    10. Lindahl, Michael & Stidsen, Thomas & Sørensen, Matias, 2019. "Quality recovering of university timetables," European Journal of Operational Research, Elsevier, vol. 276(2), pages 422-435.

    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. De Boeck, Liesje & Beliën, Jeroen & Creemers, Stefan, 2016. "A column generation approach for solving the examination-timetabling problemAuthor-Name: Woumans, Gert," European Journal of Operational Research, Elsevier, vol. 253(1), pages 178-194.
    2. Stefano Gualandi & Federico Malucelli, 2012. "Exact Solution of Graph Coloring Problems via Constraint Programming and Column Generation," INFORMS Journal on Computing, INFORMS, vol. 24(1), pages 81-100, February.
    3. Edmund Burke & Jakub Mareček & Andrew Parkes & Hana Rudová, 2012. "A branch-and-cut procedure for the Udine Course Timetabling problem," Annals of Operations Research, Springer, vol. 194(1), pages 71-87, April.
    4. Fabrizio Borghini & Isabel Méndez-Díaz & Paula Zabala, 2020. "An exact algorithm for the edge coloring by total labeling problem," Annals of Operations Research, Springer, vol. 286(1), pages 11-31, March.
    5. Natalia Castro & María A. Garrido-Vizuete & Rafael Robles & María Trinidad Villar-Liñán, 2020. "Contrast in greyscales of graphs," Journal of Combinatorial Optimization, Springer, vol. 39(3), pages 874-898, April.
    6. Alex Gliesch & Marcus Ritt, 2022. "A new heuristic for finding verifiable k-vertex-critical subgraphs," Journal of Heuristics, Springer, vol. 28(1), pages 61-91, February.
    7. Esmaeilbeigi, Rasul & Mak-Hau, Vicky & Yearwood, John & Nguyen, Vivian, 2022. "The multiphase course timetabling problem," European Journal of Operational Research, Elsevier, vol. 300(3), pages 1098-1119.
    8. Nicolas Zufferey & Olivier Labarthe & David Schindl, 2012. "Heuristics for a project management problem with incompatibility and assignment costs," Computational Optimization and Applications, Springer, vol. 51(3), pages 1231-1252, April.
    9. Andrea Bettinelli & Valentina Cacchiani & Roberto Roberti & Paolo Toth, 2015. "An overview of curriculum-based course timetabling," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 313-349, July.
    10. Song, Kwonsik & Kim, Sooyoung & Park, Moonseo & Lee, Hyun-Soo, 2017. "Energy efficiency-based course timetabling for university buildings," Energy, Elsevier, vol. 139(C), pages 394-405.
    11. R. Alan Bowman, 2021. "Developing Optimal Student Plans of Study," Interfaces, INFORMS, vol. 51(6), pages 409-421, November.
    12. Xiao-Feng Xie & Jiming Liu, 2009. "Graph coloring by multiagent fusion search," Journal of Combinatorial Optimization, Springer, vol. 18(2), pages 99-123, August.
    13. Gerhard Post & Samad Ahmadi & Sophia Daskalaki & Jeffrey Kingston & Jari Kyngas & Cimmo Nurmi & David Ranson, 2012. "An XML format for benchmarks in High School Timetabling," Annals of Operations Research, Springer, vol. 194(1), pages 385-397, April.
    14. Shen, Yunzhuang & Sun, Yuan & Li, Xiaodong & Eberhard, Andrew & Ernst, Andreas, 2023. "Adaptive solution prediction for combinatorial optimization," European Journal of Operational Research, Elsevier, vol. 309(3), pages 1392-1408.
    15. N. Cherfi & M. Hifi, 2010. "A column generation method for the multiple-choice multi-dimensional knapsack problem," Computational Optimization and Applications, Springer, vol. 46(1), pages 51-73, May.
    16. Jamie Fairbrother & Adam N. Letchford & Keith Briggs, 2018. "A two-level graph partitioning problem arising in mobile wireless communications," Computational Optimization and Applications, Springer, vol. 69(3), pages 653-676, April.
    17. Vermuyten, Hendrik & Lemmens, Stef & Marques, Inês & Beliën, Jeroen, 2016. "Developing compact course timetables with optimized student flows," European Journal of Operational Research, Elsevier, vol. 251(2), pages 651-661.
    18. Daniel Kowalczyk & Roel Leus, 2018. "A Branch-and-Price Algorithm for Parallel Machine Scheduling Using ZDDs and Generic Branching," INFORMS Journal on Computing, INFORMS, vol. 30(4), pages 768-782, November.
    19. De Causmaecker, Patrick & Demeester, Peter & Vanden Berghe, Greet, 2009. "A decomposed metaheuristic approach for a real-world university timetabling problem," European Journal of Operational Research, Elsevier, vol. 195(1), pages 307-318, May.
    20. Barry McCollum & Andrea Schaerf & Ben Paechter & Paul McMullan & Rhyd Lewis & Andrew J. Parkes & Luca Di Gaspero & Rong Qu & Edmund K. Burke, 2010. "Setting the Research Agenda in Automated Timetabling: The Second International Timetabling Competition," INFORMS Journal on Computing, INFORMS, vol. 22(1), pages 120-130, February.

    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:179:y:2010:i:1:p:105-130:10.1007/s10479-010-0716-z. 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.