A branch-and-cut procedure for the Udine Course Timetabling problem

This paper describes a branch-and-cut procedure for an extension of the bounded colouring problem, generally known as curriculum-based university course timetabling. In particular, we focus on Udine Course Timetabling [di Gaspero and Schaerf, J. Math. Model. Algorithms 5:1], which has been used in Track 3 of the 2007 International Timetabling Competition. First, we present an alternative integer programming formulation for this problem, which uses a lower than usual number of variables and a mildly-increased number of constraints (exponential in the number of periods per day). Second, we present the corresponding branch-and-cut procedure, where constraints from enumeration of event/free-period patterns, necessary to reach optimality, are added only when they are violated. We also describe further problem-specific cuts from bounds implied by the soft constraints, cuts from patterns given by days of instruction and free days, and all related separation routines. We also discuss applicability of standard cuts from graph colouring and weighted matching. The results of our preliminary experimentation with an implementation using ILOG Concert and CPLEX 10 are provided. Within 15 minutes, it is possible to find provably optimal solutions to two instances (comp01 and comp11) and good lower bounds for several other instances.