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A supernodal formulation of vertex colouring with applications in course timetabling

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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.

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Authors and Affiliations

  1. Automated Scheduling, Optimisation and Planning Group, School of Computer Science, The University of Nottingham, Nottingham, NG8 1BB, UK

    Edmund K. Burke, Jakub Mareček & Andrew J. Parkes

  2. Faculty of Informatics, Masaryk University, Botanická 68a, Brno, 602 00, Czech Republic

    Jakub Mareček & Hana Rudová

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  1. Edmund K. Burke
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  2. Jakub Mareček
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Correspondence to Jakub Mareček.

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Burke, E.K., Mareček, J., Parkes, A.J. et al. A supernodal formulation of vertex colouring with applications in course timetabling. Ann Oper Res 179, 105–130 (2010). https://doi.org/10.1007/s10479-010-0716-z

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