Decomposition of university course timetabling

A systematic study of subproblems and their complexities
  • Britta Herres
  • Heinz SchmitzEmail author
S.I.: PATAT 2018


Suppose we like to find non-overlapping periods for a set of events which may have multiple teachers assigned, is this easy or hard in terms of complexity? Or assume that only a single teacher is fixed per event, but we like to allocate rooms and periods simultaneously. What if a single teacher and a room is already given and we look for periods alone? And how do requests of teachers for specific rooms or additional student conflicts change the computational complexities of these questions from university course timetabling (UCT)? We provide a complete hard/easy-list of all UCT subproblems derived from a typical set of hard constraints. We obtain this list with a systematic study of the fine structure of UCT in terms of complexity w.r.t. the order in which rooms, periods and teachers are assigned to events. These kind of subproblems appear in practice when some entities in a timetable are fixed while the assignments of others are (re-)computed, and they also appear as necessary conditions for the existence of feasible timetables. Moreover, we identify which of the seemingly different subproblems are essentially the same computational tasks by reducing them to the same bipartite assignment problem, and we discuss some variations of constraints.


Foundations of university course timetabling Complexity analysis Decomposition approach Bipartite assignment problems 



We are very grateful to our anonymous reviewers for their very valuable comments.


  1. Asratian, A., & de Werra, D. (2002). A generalized class-teacher model for some timetabling problems. European Journal of Operational Research, 143(3), 531–542.CrossRefGoogle Scholar
  2. Babaei, H., Karimpour, J., & Hadidi, A. (2015). A survey of approaches for university course timetabling problem. Computers and Industrial Engineering, 86, 43–59.CrossRefGoogle Scholar
  3. Chen, H. (2006). Logic column 17: A rendezvous of logic, complexity, and algebra. CoRR abs/cs/0611018, arXiv:cs/0611018 CrossRefGoogle Scholar
  4. Cole, R., Ost, K., & Schirra, S. (2001). Edge-coloring bipartite multigraphs in O (E log D) time. Combinatorica, 21(1), 5–12.CrossRefGoogle Scholar
  5. Cooper, T. B., & Kingston, J. H. (1996). The complexity of timetable construction problems. In E. Burke & P. Ross (Eds.), Practice and theory of automated timetabling (pp. 281–295). Heidelberg: Springer.CrossRefGoogle Scholar
  6. Csima, J. (1965). Investigations on a time-table problem. Ph.D. thesis, School of Graduate Studies, University of Toronto.Google Scholar
  7. de Werra, D. (1971). Construction of school timetables by flow methods. INFOR Journal, 9(1), 12–22.Google Scholar
  8. de Werra, D. (2003). Constraints of availability in timetabling and scheduling. In E. Burke & P. De Causmaecker (Eds.), Practice and theory of automated timetabling IV (pp. 3–23). Heidelberg: Springer.CrossRefGoogle Scholar
  9. Dostert, M., Politz, A., & Schmitz, H. (2016). A complexity analysis and an algorithmic approach to student sectioning in existing timetables. Journal of Scheduling, 19(3), 285–293.CrossRefGoogle Scholar
  10. Even, S., Itai, A., & Shamir, A. (1975). On the complexity of time table and multi-commodity flow problems. In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, Washington, DC, SFCS ’75, pp. 184–193.Google Scholar
  11. Ford, L. R., & Fulkerson, D. R. (1987). Maximal flow through a network (pp. 243–248). Boston: Birkhäuser Boston.Google Scholar
  12. Garey, M. R., & Johnson, D. S. (1990). Computers and intractability: A guide to the theory of NP-completeness. New York: W. H. Freeman and Co.Google Scholar
  13. Gaspero, L.D., Mccollum, B., & Schaerf, A. (2007). The second international timetabling competition (itc-2007): Curriculum-based course timetabling (track 3. Tech. rep.Google Scholar
  14. Gotlieb, C.C. (1962). The Construction of class-teacher time-tables. In: IFIP Congress, pp 73–77Google Scholar
  15. Holyer, I. (1981). The NP-completeness of edge-coloring. SIAM Journal on Computing, 10(4), 718–720.CrossRefGoogle Scholar
  16. Hopcroft, J. E., & Karp, R. M. (1973). An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing, 2(4), 225–231.CrossRefGoogle Scholar
  17. Karp, R. M. (1972). Reducibility among combinatorial problems (pp. 85–103). Boston: Springer.Google Scholar
  18. Kingston, J. H. (2014). Timetable construction: the algorithms and complexity perspective. Annals of Operations Research, 218(1), 249–259.CrossRefGoogle Scholar
  19. Kostuch, P. (2005). The university course timetabling problem with a three-phase approach. In E. Burke & M. Trick (Eds.), Practice and theory of automated timetabling V (pp. 109–125). Heidelberg: Springer.CrossRefGoogle Scholar
  20. Kristiansen, S., & Stidsen, T. (2013). A comprehensive study of educational timetabling—a survey. Report 8.2013, Department of Management Engineering, Technical University of DenmarkGoogle Scholar
  21. Lach, G., & Lübbecke, M. E. (2012). Curriculum based course timetabling: new solutions to Udine benchmark instances. Annals of Operations Research, 194(1), 255–272.CrossRefGoogle Scholar
  22. Lewis, R., Paechter, B., & Mccollum, B. (2007). Post enrolment based course timetabling: A description of the problem model used for track two of the second international timetabling competition. Cardiff University, Cardiff Business School, Accounting and Finance Section, Cardiff Accounting and Finance Working PapersGoogle Scholar
  23. Marx, D. (2005). NP-completeness of list coloring and precoloring extension on the edges of planar graphs. Journal of Graph Theory, 49(4), 313–324.CrossRefGoogle Scholar
  24. McCollum, B. (2006). University Timetabling: Bridging the Gap between Research and Practice. In: in Proceedings of the 5th International Conference on the Practice and Theory of Automated Timetabling, Springer, pp. 15–35.Google Scholar
  25. Müller, T., Rudova, H., & Müllerova, Z. (2018). University course timetabling and international timetabling competition 2019. In: Proceedings of the 12th International Conference of the Practice and Theory of Automated Timetabling (PATAT 2018), Vienna, Austria, pp. 5 – 31.Google Scholar
  26. Paechter, B., Gambardella, L.M., & Rossi-Doria, O. (2002). The first international timetabling competition. URL
  27. Plaisted, D. A., & Zaks, S. (1980). An NP-complete matching problem. Discrete Applied Mathematics, 2(1), 65–72.CrossRefGoogle Scholar
  28. Rudova, H. (2015). University course timetabling—from theory to practice. In: Multidisciplinary International Scheduling Conference (MISTA 2015) (Talk), Prague, Czech Republic, URL
  29. Schaefer, T.J. (1978). The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1-3, 1978, San Diego, California, USA, pp. 216–226,
  30. Schindl, D. (2019). Optimal student sectioning on mandatory courses with various sections numbers. Annals of Operations Research, 275(1), 209–221. Scholar
  31. Tanimoto, S. L., Itai, A., & Rodeh, M. (1978). Some matching problems for bipartite graphs. Journal of the ACM, 25(4), 517–525.CrossRefGoogle Scholar
  32. ten Eikelder, H. M. M., & Willemen, R. J. (2001). Some complexity aspects of secondary school timetabling problems. In E. Burke & W. Erben (Eds.), Practice and theory of automated timetabling III (pp. 18–27). Heidelberg: Springer.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Hochschule TrierTrierGermany

Personalised recommendations