A MAX -MIN Ant System for the University Course Timetabling Problem Krzysztof Socha, Joshua Knowles, and Michael Sampels IRIDIA, Universit´e Libre de Bruxelles, CP 194/6, Av. Franklin D. Roosevelt 50, 1050 Bruxelles, Belgium {ksocha|jknowles|msampels}@ulb.ac.be http://iridia.ulb.ac.be Abstract. We consider a simplification of a typical university course timetabling problem involving three types of hard and three types of soft constraints. A MAX -MIN Ant System, which makes use of a separate local search routine, is proposed for tackling this problem. We devise an appropriate construction graph and pheromone matrix representation after considering alternatives. The resulting algorithm is tested over a set of eleven instances from three classes of the problem. The results demonstrate that the ant system is able to construct significantly better timetables than an algorithm that iterates the local search procedure from random starting solutions. 1 Introduction Course timetabling problems are periodically faced by virtually every school, college and university in the world. In a basic problem, a set of times must be assigned to a set of events (e.g., classes, lectures, tutorials, etc.) in such a way that all of the students can attend all of their respective events. Some pairs of events are edge-constrained (e.g., some students must attend both events), so that they must be scheduled at different times, and this yields what is essentially a form of vertex colouring problem. In addition, real timetables must usually satisfy a large and diverse array of supplementary constraints which are difficult to describe in a generic manner. However, the general university course timetabling problem (UCTP) is known to be NP-hard, as are many of the subproblems associated with additional constraints [5, 9, 22]. Of course, the difficulty of any particular instance of the UCTP depends on many factors and, while little is known about how to estimate difficulty, it seems that the assignment of rooms makes the problem significantly harder than vertex colouring, in general. Current methods for tackling timetabling problems include evolutionary algo- rithms [6], simulated annealing [13] and tabu search [14]. Many problem-specific heuristics also exist for timetabling and its associated sub-problems. These have been used within evolutionary methods and other generic search methods, ei- ther as ‘hyper-heuristics’ [1, 7], or to repair or decode indirect solution repre- sentations [16]. Local search has also been used successfully within a memetic algorithm to do real-world exam timetabling [3]. Although several ant colony