A supernodal formulation of vertex colouring with applications in course timetabling

Abstract

For many problems in Scheduling and Timetabling the choice of an 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 formulation using "supernodes". In the definition of George and McIntyre [SIAM J. Numer. Anal. 15 (1978), no. 1, 90-112], "supernode" is a complete subgraph, where each two vertices have the same neighbourhood outside of the subgraph. Seen another way, the algorithm for obtaining the best possible partition of an arbitrary graph into supernodes, which we give and show to be polynomial-time, makes it possible to use any formulation of vertex multicolouring to encode vertex colouring. The power of this approach is shown on the benchmark problem of Udine Course Timetabling. Results from empirical tests on DIMACS colouring instances, in addition to instances from other timetabling applications, are also provided and discussed.

The Interleaved Constructive Memetic Algorithm and its application to timetabling

“…As a potential example, the structure of the PSTP is similar to those used in the ITC2007 course timetabling track, [35], and so this suggests that ICMA be also tested on this domain. We note that [55] uses an integer programming method (as opposed to a metaheuristic) but that it does exploit the given grouping structure of the events in the problem; thus re-iterating the general potential utility of structure exploitation.…”

Operational Research in education

“…The sequential approach therefore assigns events (examinations or lessons) one by one, starting with those considered the most difficult to schedule, attempting to avoid violating the hard constraints at each stage. Various heuristics have been developed to solve the scheduling problem , and there are many examples of their application to course and examination scheduling (Romero 1982;Balakrishnan 1991;Hertz 1998;Carter and Johnson 2001;Burke and Newall 2004;Burke et al 2010c).…”

Comments on: An overview of curriculum-based course timetabling

“…The paper discusses about instance generators, and in particular the ones developed by Mareček (see Burke et al 2010) and by Lopes (see Lopes and Smith-Miles 2010), which are available on the web. I would like to mention our experience with the latter one, which has been very positive.…”