A supernodal formulation of vertex colouring with applications in course timetabling

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Authors	BURKE K MAREČEK Jakub PARKES Andrew RUDOVÁ Hana
Year of publication	2010
Type	Article in Periodical
Magazine / Source	Annals of Operations Research
MU Faculty or unit	Faculty of Informatics
Citation
web	DOI pre-print
Field	Applied statistics, operation research
Keywords	Vertex colouring; Graph colouring; Multicolouring; Supernode; Module; Integer programming
Description	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.
Related projects:	Highly Parallel and Distributed Computing Systems Dynamic Aspects of Scheduling