Penalising Patterns in Timetables: Strengthened Integer Programming Formulations

Many complex timetabling problems, such as university course timetabling [1, 2] and employee rostering [3], have an underpinning bounded graph colouring component, a pattern penalisation component and a number of side constraints. The bounded graph colouring component corresponds to hard constraints such as “each student attends all events of courses of his choice, no student can be in two rooms at the same time, and there can be at most a given number of events taking place during each period”. Despite the intractability of bounded graph colouring, it is often easy to come up with feasible solutions for instances with hundreds of events and hundreds of distinct enrollments. Much more challenging are requirements such as “students should not have gaps in their individual daily timetables”, which often make the problem over-constrained. The key to tackling this challenge is a suitable formulation of “soft” constraints counting and minimising penalties incurred by matches of various patterns. Several integer programming formulations of such pattern penalising constraints are studied in this paper.