On a Clique-Based Integer Programming Formulation of Vertex Colouring with Applications in Course Timetabling

TL;DR

This paper introduces a new clique-based integer programming formulation for vertex colouring, tailored for course timetabling, which can handle soft constraints and shows promising empirical results.

Contribution

It presents a novel vertex colouring formulation using clique partitions that is adaptable to complex real-world timetabling constraints.

Findings

Empirical results on Udine Course Timetabling are encouraging.

The formulation can incorporate soft constraints easily.

Performance depends on the quality of the clique partition.

Abstract

Vertex colouring is a well-known problem in combinatorial optimisation, whose alternative integer programming formulations have recently attracted considerable attention. This paper briefly surveys seven known formulations of vertex colouring and introduces a formulation of vertex colouring using a suitable clique partition of the graph. This formulation is applicable in timetabling applications, where such a clique partition of the conflict graph is given implicitly. In contrast with some alternatives, the presented formulation can also be easily extended to accommodate complex performance indicators (``soft constraints'') imposed in a number of real-life course timetabling applications. Its performance depends on the quality of the clique partition, but encouraging empirical results for the Udine Course Timetabling problem are reported.