Skip to main content

The University Course Timetabling Problem

  • Chapter
  • First Online:
Fairness in Academic Course Timetabling

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 678))

  • 740 Accesses

Abstract

Despite its simplicity, the UCTP is a computationally tough problem. We will review solution approaches to the UCTP and related problems and investigate combinatorial properties of the UCTP search space. We focus on establishing conditions that guarantee the connectedness of all clash-free timetables with respect to the Kempe-exchange operation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. Achá, R.A., Nieuwenhuis, R.: Curriculum-based course timetabling with SAT and MaxSAT. Ann. Oper. Res. 1–21 (2012). doi:10.1007/s10479-012-1081-x

  2. Achá, R.A., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E.: Cardinality networks and their applications. In: Kullmann, O. (ed.) 12th International Conference on Theory and Applications of Satisfiability Testing, SAT’09. Lecture Notes in Computer Science, vol. 5584, pp. 167–180. Springer, Berlin (2009)

    Google Scholar 

  3. Arntzen, H., Løkketangen, A.: A tabu search heuristic for a university timetabling problem. In: Ibaraki, T., Nonobe, K., Yagiura, M. (eds.) Metaheuristics: Progress as Real Problem Solvers. Operations Research/Computer Science Interfaces Series, vol. 32, pp. 65–85. Springer, New York (2005). doi:10.1007/0-387-25383-1_3

    Chapter  Google Scholar 

  4. Burke, E.K., Bykov, Y.: The late acceptance hill-climbing heuristic. Technical Report, CSM-192, Computing Science and Mathematics, University of Stirling (2012)

    Google Scholar 

  5. Baker, B.S., Coffman, Jr., E.G.: Mutual exclusion scheduling. Theor. Comput. Sci. 162(2), 225–243 (1996). doi:10.1016/0304-3975(96)00031-X

    Article  Google Scholar 

  6. Bonsma, P., Cereceda, L.: Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theor. Comput. Sci. 410, 5215–5226 (2009). doi:10.1016/j.tcs.2009.08.023

    Article  Google Scholar 

  7. Burke, E.K., de Werra, D., Kingston, J.H.: Applications to timetabling. In: Gross, J.L., Yellen, J. (eds.) Handbook of Graph Theory, 1st edn., Chap. 5.6, pp. 475–483. CRC Press, Boca Raton (2003)

    Google Scholar 

  8. Burke, E.K., Eckersley, A.J., McCollum, B., Petrovic, S., Qu, R.: Hybrid variable neighbourhood approaches to university exam timetabling. Eur. J. Oper. Res. 206(1), 46–53 (2010). doi:10.1016/j.ejor.2010.01.044

    Article  Google Scholar 

  9. Bäck, T., Fogel, D.B., Michalewicz, Z. (eds.): Handbook of Evolutionary Computation, 1st edn. IOP Publishing, Bristol (1997)

    Google Scholar 

  10. Birattari, M.: The Race Package. http://cran.r-project.org/web/packages/race (2013). Accessed Sept 2013

  11. Burke, E.K., Silva, J.D.L., Soubeiga, E.: Multi-objective hyper-heuristic approaches for space allocation and timetabling. In: Ibaraki, T., Nonobe, K., Yagiura, M. (eds.) Metaheuristics: Progress as Real Problem Solvers. Operations Research/Computer Science Interfaces Series, vol. 32, pp. 129–158. Springer, New York (2005). doi:10.1007/0-387-25383-1_6

    Chapter  Google Scholar 

  12. Bonamy, M., Johnson, M., Lignos, I., Patel, V., Paulusma, D.: On the diameter of reconfiguration graphs for vertex colourings. Electron. Notes Discrete Math. 38(1), 161–166 (2011). doi:10.1016/j.endm.2011.09.028

    Article  Google Scholar 

  13. Bonamy, M., Johnson, M., Lignos, I., Patel, V., Paulusma, D.: Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs. J. Comb. Optim. 1–12 (2012). doi:10.1007/s10878-012-9490-y

  14. Burke, E.K., McCollum, B., Meisels, A., Petrovic, S., Qu, R.: A graph-based hyper-heuristic for educational timetabling problems. Eur. J. Oper. Res. 176(1), 177–192 (2007)

    Article  Google Scholar 

  15. Burke, E.K., Newall, J.P.: A multistage evolutionary algorithm for the timetable problem. IEEE Trans. Evol. Comput. 3(1), 63–74 (1999). doi:10.1109/4235.752921

    Article  Google Scholar 

  16. Bonsma, P.: The complexity of rerouting shortest paths. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) Mathematical Foundations of Computer Science 2012. Lecture Notes in Computer Science, vol. 7464, pp. 222–233. Springer, Berlin/Heidelberg (2012). doi:10.1007/978-3-642-32589-2_22

    Chapter  Google Scholar 

  17. Birattari, M., Yuan, Z., Balaprakash, P., Stützle, T.: F-race and iterated f-race: an overview. In: Bartz-Beielstein, T., Chiarandini, M., Paquete, L., Preuss, M. (eds.) Experimental Methods for the Analysis of Optimization Algorithms, pp. 311–336. Springer, Berlin/Heidelberg (2010). doi:10.1007/978-3-642-02538-9_13

    Chapter  Google Scholar 

  18. Batagelj, V., Zaversnik, M.: An O(m) algorithm for cores decomposition of networks. CoRR cs.DS/0310049 (2003)

    Google Scholar 

  19. Carter, M.W.: A comprehensive course timetabling and student scheduling system at the university of Waterloo. In: Selected Papers from the Third International Conference on Practice and Theory of Automated Timetabling III, PATAT ’00, pp. 64–84. Springer, London (2001)

    Google Scholar 

  20. Chiarandini, M., Gaspero, L.D., Gualandi, S., Schaerf, A.: The balanced academic curriculum problem revisited. J. Heuristics 18(1), 119–148 (2012)

    Article  Google Scholar 

  21. Ceschia, S., Gaspero, L.D., Schaerf, A.: Design, engineering, and experimental analysis of a simulated annealing approach to the post-enrolment course timetabling problem. Comput. Oper. Res. 39(7), 1615–1624 (2012). doi:10.1016/j.cor.2011.09.014

    Article  Google Scholar 

  22. Chen, J.: A new SAT encoding of the at-most-one constraint. In: Proceedings of Constraint Modelling and Reformulation (2010)

    Google Scholar 

  23. Cooper, T.B., Kingston, J.H.: The complexity of timetable construction problems. In: Practice and Theory of Automated Timetabling, pp. 283–295 (1995). doi:10.1007/3-540-61794-9_66

  24. Carter, M.W., Laporte, G.: Recent developments in practical examination timetabling. In: Burke, E.K., Ross, P. (eds.) Practice and Theory of Automated Timetabling. Lecture Notes in Computer Science, vol. 1153, pp. 1–21. Springer, Berlin/Heidelberg (1996). doi:10.1007/3-540-61794-9_49

    Chapter  Google Scholar 

  25. Cioppa, T.M., Lucas, T.W.: Efficient nearly orthogonal and space-filling latin hypercubes. Technometrics 49(1), 45–55 (2007)

    Article  Google Scholar 

  26. Carter, M.W., Laporte, G., Lee, S.Y.: Examination timetabling: algorithmic strategies and applications. J. Oper. Res. Soc. 47(3), 373–383 (1996)

    Article  Google Scholar 

  27. Castro, C., Manzano, S.: Variable and value ordering when solving balanced academic curriculum problems. In: 6th Workshop of the ERCIM WG on Constraints (2001)

    Google Scholar 

  28. Conover, W.J.: Practical Nonparametric Statistics. Wiley Series in Probability and Statistics, 3rd edn. Wiley, New York (2006)

    Google Scholar 

  29. Corneil, D.G.: Lexicographic breadth first search—a survey. In: Hromkovic, J., Nagl, M., Westfechtel, B. (eds.) Graph-Theoretic Concepts in Computer Science. Lecture Notes in Computer Science, vol. 3353, pp. 1–19. Springer, Berlin/Heidelberg (2005). doi:10.1007/978-3-540-30559-0_1

    Chapter  Google Scholar 

  30. Cereceda, L., van den Heuvel, J., Johnson, M.: Finding paths between 3-colorings. J. Graph Theory 67(1), 69–82 (2011). doi:10.1002/jgt.20514

    Article  Google Scholar 

  31. Di Gaspero, L., McCollum, B., Schaerf, A.: The second international timetabling competition (ITC-2007): curriculum-based course timetabling (Track 3). In: Proceedings of the 1st International Workshop on Scheduling, a Scheduling Competition (SSC) (2007)

    Google Scholar 

  32. Di Gaspero, L., Schaerf, A.: Curriculum-based course timetabling web-site. http://satt.diegm.uniud.it/ctt/ (2013). Accessed Sept 2013

  33. Di Gaspero, L., Schaerf, A.: Tabu search techniques for examination timetabling. In: Burke, E., Erben, W. (eds.) Practice and Theory of Automated Timetabling III. Lecture Notes in Computer Science, vol. 2079, pp. 104–117. Springer, Berlin/Heidelberg (2001)

    Chapter  Google Scholar 

  34. Di Gaspero, L., Schaerf, A.: Neighborhood portfolio approach for local search applied to timetabling problems. J. Math. Model. Algorithm 5(1), 65–89 (2006). doi:10.1007/s10852-005-9032-z

    Article  Google Scholar 

  35. Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Commun. ACM 5(7), 394–397 (1962). doi:10.1145/368273.368557

    Article  Google Scholar 

  36. Dostert, M., Politz, A., Schmitz, H.: Algorithms and complexity of student sectioning for existing timetables. In: Proceedings of the 6th Multidisciplinary International Conference on Scheduling: Theory and Applications (MISTA), pp. 218–229 (2013)

    Google Scholar 

  37. Dorigo, M., Stützle, T.: Ant Colony Optimization. MIT Press, Cambridge (2004)

    Book  Google Scholar 

  38. Dueck, G.: New optimization heuristics: the great deluge algorithm and the record-to-record travel. J. Comput. Phys. 104(1), 86–92 (1993). doi:10.1006/jcph.1993.1010

    Article  Google Scholar 

  39. de Werra, D.: An introduction to timetabling. Eur. J. Oper. Res. 19(2), 151–162 (1985)

    Article  Google Scholar 

  40. de Werra, D.: Restricted coloring models for timetabling. Discrete Math. 165–166(15), 161–170 (1997). doi:10.1016/S0012-365X(96)00208-7

    Article  Google Scholar 

  41. Even, S., Itai, A., Shamir, A.: On the complexity of timetable and multicommodity flow problems. SIAM J. Comput. 5(4), 691–703 (1976). doi:10.1137/0205048

    Article  Google Scholar 

  42. Erben, W.: A grouping genetic algorithm for graph colouring and exam timetabling. In: Burke, E.K., Erben, W. (eds.) Practice and Theory of Automated Timetabling III. Lecture Notes in Computer Science, vol. 2079, pp. 132–156. Springer, Berlin/Heidelberg (2001). doi:10.1007/3-540-44629-X_9

    Chapter  Google Scholar 

  43. Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) Theory and Applications of Satisfiability Testing. Lecture Notes in Computer Science, vol. 2919, pp. 502–518. Springer, Berlin/Heidelberg (2004). doi:10.1007/978-3-540-24605-3_37

    Chapter  Google Scholar 

  44. Falkenauer, E.: A new representation and operators for genetic algorithms applied to grouping problems. Evol. Comput. 2(2), 123–144 (1994)

    Article  Google Scholar 

  45. Falkenauer, E.: Genetic Algorithms and Grouping Problems. Wiley, New York (1998)

    Google Scholar 

  46. Frisch, A.M., Giannaros, P.A.: SAT encodings of the at-most-k constraint. In: Proceedings of International Workshop on Modelling and Reformulating Constraint Satisfaction Problems (2010)

    Google Scholar 

  47. Friedman, M.: The use of ranks to avoid the assumption of normality implicit in the analysis of variance. J. Am. Stat. Assoc. 32(200), 675–701 (1937). doi:10.1080/01621459.1937.10503522

    Article  Google Scholar 

  48. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)

    Google Scholar 

  49. Gopalan, P., Kolaitis, P.G., Maneva, E., Papadimitriou, C.H.: The connectivity of boolean satisfiability: computational and structural dichotomies. SIAM J. Comput. 38(6), 2330–2355 (2009). doi:10.1137/07070440X

    Article  Google Scholar 

  50. Glover, F., Laguna, M.: Tabu Search. Kluwer Academic Publishers, Norwell (1997)

    Book  Google Scholar 

  51. Glover, F.: Tabu search: part I. ORSA J. Comput 1(3), 190–206 (1989)

    Article  Google Scholar 

  52. Hearn, R.A., Demaine, E.D.: PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theor. Comput. Sci. 343(1–2), 72–96 (2005). doi:10.1016/j.tcs.2005.05.008

    Article  Google Scholar 

  53. Hnich, B., Kiziltan, Z., Walsh, T.: Modelling a balanced academic curriculum problem. In: Proceedings of CP-AI-OR-2002, pp. 121–131 (2002)

    Google Scholar 

  54. Ito, T., Demaine, E.D., Harvey, N.J.A., Papadimitriou, C.H., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theor. Comput. Sci. 412(12–14), 1054–1065 (2011)

    Article  Google Scholar 

  55. Jerrum, M.: A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Struct. Algorithm 7(2), 157–165 (1995). doi:10.1002/rsa.3240070205

    Article  Google Scholar 

  56. Johnson, D.S.: A theoretician’s guide to the experimental analysis of algorithms. In: Data Structures, Near Neighbor Searches, and Methodology: Fifth and Sixth DIMACS Implementation Challenges, vol. 59, pp. 215–250. American Mathematical Society, Providence (2002)

    Google Scholar 

  57. Kennedy, J.: Particle swarm optimization. In: Encyclopedia of Machine Learning, pp. 760–766. Springer, New York (2010)

    Google Scholar 

  58. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)

    Article  Google Scholar 

  59. Kőnig, D.: Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math. Ann. 77(4), 453–465 (1916). doi:10.1007/BF01456961

    Article  Google Scholar 

  60. Kostuch, P.: The university course timetabling problem with a three-phase approach. In: Burke, E.K., Trick, M. (eds.) Practice and Theory of Automated Timetabling. Lecture Notes in Computer Science, vol. 3616, pp. 109–125. Springer, Berlin/Heidelberg (2005). doi:10.1007/11593577_7

    Chapter  Google Scholar 

  61. Lewis, R.: Metaheuristics for university course timetabling. Ph.D. thesis, Napier University, Edinburgh (2006)

    Google Scholar 

  62. Lewis, R.: A survey of metaheuristic-based techniques for university timetabling problems. OR Spectrum 30(1), 167–190 (2008). doi:10.1007/s00291-007-0097-0

    Article  Google Scholar 

  63. Lü, Z., Hao, J.-K.: Adaptive tabu search for course timetabling. Eur. J. Oper. Res. 200(1), 235–244 (2010). doi:10.1016/j.ejor.2008.12.007

    Article  Google Scholar 

  64. López-Ibáñez, M., Dubois-Lacoste, J., Stützle, T., Birattari, M.: The irace package, iterated race for automatic algorithm configuration. Technical Report, TR/IRIDIA/2011-004, IRIDIA, Université Libre de Bruxelles (2011)

    Google Scholar 

  65. Lewis, R., Paechter, B.: GGA experimental results. http://www.soc.napier.ac.uk/~benp/centre/timetabling/experimentalresults2.htm (2013). Accessed Sept 2013

  66. Lewis, R., Paechter, B.: New “harder” instances for the university course timetabling problem. http://www.soc.napier.ac.uk/~benp/centre/timetabling/harderinstances.htm (2013). Accessed Sept 2013

  67. Lewis, R., Paechter, B.: New crossover operators for timetabling with evolutionary algorithms. In: Proceedings of the 5th International Conference on Recent Advances in Soft Computing (RASC), pp. 189–195 (2004)

    Google Scholar 

  68. Lewis, R., Paechter, B.: Finding feasible timetables using group-based operators. IEEE Trans. Evol. Comput. 11(3), 397–413 (2007)

    Article  Google Scholar 

  69. Lucarelli, G.: Scheduling in computer and communication systems and generalized graph coloring problems. Ph.D. thesis, Athens University Economics and Business (AUEB) (2009)

    Google Scholar 

  70. Las Vergnas, M., Meyniel, H.: Kempe classes and the Hadwiger conjecture. J. Combin. Theory Ser. B 31(1), 95–104 (1981). doi:10.1016/S0095-8956(81)80014-7

    Article  Google Scholar 

  71. Liu, Y., Zhang, D., Chin, F.Y.L.: A clique-based algorithm for constructing feasible timetables. Optim. Methods Software 26(2), 281–294 (2011). doi:10.1080/10556781003664739

    Article  Google Scholar 

  72. Matula, D.W.: A min-max theorem for graphs with application to graph coloring. SIAM Rev. 10(4), 467–490 (1968). doi:10.1137/1010115

    Article  Google Scholar 

  73. Merlot, L.T.G., Boland, N., Hughes, B.D., Stuckey, P.J.: A hybrid algorithm for the examination timetabling problem. In: Proceedings of the 4th International Conference on the Practice and Theory of Automated Timetabling (PATAT), pp. 207–231. Springer, Heidelberg (2002). doi:10.1007/978-3-540-45157-0_14

  74. Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Proceedings of the 38th Annual Design Automation Conference, pp. 530–535. ACM, New York (2001)

    Google Scholar 

  75. Mohar, B.: Kempe equivalence of colorings. In: Bondy, A., Fonlupt, J., Fouquet, J.-L., Fournier, J.-C., Ramirez Alfonsin, J.L. (eds.) Graph Theory in Paris. Trends in Mathematics, pp. 287–297. Birkhäuser, Basel (2007). doi:10.1007/978-3-7643-7400-6_22

    Chapter  Google Scholar 

  76. McCollum, B., Schaerf, A., Paechter, B., McMullan, P., Lewis, R., Parkes, A.J., Gaspero, L.D., Qu, R., Burke, E.K.: Setting the research agenda in automated timetabling: the second international timetabling competition. INFORMS J. Comput. 22(1), 120–130 (2010)

    Article  Google Scholar 

  77. Nothegger, C., Mayer, A., Chwatal, A., Raidl, G.R.: Solving the post enrolment course timetabling problem by ant colony optimization. Ann. Oper. Res. 194(1), 325–339 (2012). doi:10.1007/s10479-012-1078-5

    Article  Google Scholar 

  78. Östergård, P.R.J.: A fast algorithm for the maximum clique problem. Discrete Appl. Math. 120(1), 197–207 (2002)

    Article  Google Scholar 

  79. Paechter, B., Gambardella, L.M., Rossi-Doria, O.: International timetabling competition 2002. http://www.idsia.ch/Files/ttcomp2002 (2013). Accessed Oct 2013

  80. Pillay, N.: A survey of school timetabling research. Ann. Oper. Res. 1–33 (2013). doi:10.1007/s10479-013-1321-8

  81. Qu, R., Burke, E.K.: Hybridizations within a graph-based hyper-heuristic framework for university timetabling problems. J. Oper. Res. Soc. 60(9), 1273–1285 (2008)

    Article  Google Scholar 

  82. Qu, R., Burke, E.K., McCollum, B., Merlot, L.T.G., Lee, S.Y.: A survey of search methodologies and automated system development for examination timetabling. J. Sched. 12(1), 55–89 (2009). doi:10.1007/s10951-008-0077-5

    Article  Google Scholar 

  83. R Development Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2008). ISBN 3-900051-07-0

    Google Scholar 

  84. Rudová, H., Müller, T., Murray, K.: Complex university course timetabling. J. Sched. 14(2), 187–207 (2011). doi:10.1007/s10951-010-0171-3

    Article  Google Scholar 

  85. Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976). doi:10.1137/0205021

    Article  Google Scholar 

  86. Schaerf, A.: A survey of automated timetabling. Artif. Intell. Rev. 13(2), 87–127 (1999). doi:10.1023/A:1006576209967

    Article  Google Scholar 

  87. Schimmelpfeng, K., Helber, S.: Application of a real-world university-course timetabling model solved by integer programming. OR Spectrum 29(4), 783–803 (2007)

    Article  Google Scholar 

  88. Sinz, C.: Towards an optimal CNF encoding of boolean cardinality constraints. In: Proceedings of the 11th International Conference on Principles and Practice of Constraint Programming (CP 2005), Sitges, pp. 827–831 (2005)

    Google Scholar 

  89. Szekeres, G., Wilf, H.S.: An inequality for the chromatic number of a graph. J. Combin. Theory 4(1), 1–3 (1968). doi:10.1016/S0021-9800(68)80081-X

    Article  Google Scholar 

  90. Tuga, M., Berretta, R., Mendes, A.: A hybrid simulated annealing with Kempe chain neighborhood for the university timetabling problem. In: Proceedings 6th ACIS International Conference on Computer and Information Science (ACIS-ICIS), pp. 400–405 (2007)

    Google Scholar 

  91. Thompson, J.M., Dowsland, K.A.: A robust simulated annealing based examination timetabling system. Comput. Oper. Res. 25(7–8), 637–648 (1998). doi:10.1016/S0305-0548(97)00101-9

    Article  Google Scholar 

  92. Uehara, R.: Linear time algorithms on chordal bipartite and strongly chordal graphs. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds.) Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 2380, pp. 993–1004. Springer, Berlin/Heidelberg (2002). doi:10.1007/3-540-45465-9_85

    Chapter  Google Scholar 

  93. Van Gelder, A.: Another look at graph coloring via propositional satisfiability. Discrete Appl. Math. 156(2), 230–243 (2008)

    Article  Google Scholar 

  94. Wilcoxon, F.: Individual comparisons by ranking methods. Biometrics Bull. 1(6), 80–83 (1945). doi:10.2307/3001968

    Article  Google Scholar 

  95. Welsh, D.J.A., Powell, M.B.: An upper bound for the chromatic number of a graph and its application to timetabling problems. Comput. J. 10(1), 85–86 (1967). doi:10.1093/comjnl/10.1.85

    Article  Google Scholar 

  96. Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(6), 103–128 (2007). doi:10.4086/toc.2007.v003a006

    Article  Google Scholar 

  97. Mühlenthaler, M., Wanka, R.: A novel event insertion heuristic for finding feasible solutions of course timetabling problems. In: Proceedings of 8th International Conferene on the Practice and Theory of Automated Timetabling (PATAT), pp. 294–304 (2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Mühlenthaler, M. (2015). The University Course Timetabling Problem. In: Fairness in Academic Course Timetabling. Lecture Notes in Economics and Mathematical Systems, vol 678. Springer, Cham. https://doi.org/10.1007/978-3-319-12799-6_2

Download citation

Publish with us

Policies and ethics