A Hybrid Simulated Annealing with Kempe Chain Neighborhood for the University Timetabling Problem Mauritsius Tuga, Regina Berretta and Alexandre Mendes School of Electrical Engineering and Computer Science The University of Newcastle, Callaghan, 2308, NSW, Australia Moris@cs.newcastle.edu.au, {Regina.Berretta,Alexandre.Mendes}@newcastle.edu.au Abstract This paper addresses the problem of finding a feasi- ble solution for the University Course Timetabling Problem (UCTP), i.e. a solution that satisfies all the so-called hard constraints. The problem is reformulated through relaxing one of its hard constraints and then creating a soft con- straint to address the relaxed constraint. The relaxed prob- lem is solved in two steps. First, a graph-based heuristic is used to construct a feasible solution of the relaxed prob- lem, and then, a Simulated Annealing (SA)-based approach is utilized to minimize the violation of the soft constraint. In order to strengthen the diversification ability of the method in the SA phase, a heuristic based on Kempe Chain neigh- borhood is embedded into the standard approach. This strategy is tested on a well-known data set, and the results are very competitive compared to the current state of the art of the UCTP. 1. Introduction The University Course Timetabling Problem (UCTP) is a well-known NP-Hard combinatorial problem. It is defined as the assignment of a set of main academic events related to a course, such as lectures, tutorials or lab sessions, to re- sources (timeslots and rooms) subject to a set of constraints. In general the set of constraints can be categorized as hard and soft. Hard constraints are those that are compulsory to be fulfilled. A timetable will not be acceptable if any hard constraint is violated. Soft constraints include some non-compulsory requirements. Even though they could be violated there is a strong demand to minimize such viola- tions. A timetable without any hard constraints violations will be referred to as a feasible timetable. The quality of a feasible timetable will be measured by the extension of its soft constraints violations. Requirements for a feasible timetable, as well as its soft constraints, differ from one university to another, as each university has different rules. However, there might be some common requirements for a timetable to be consid- ered feasible. They are for instance, no students, as well as lecturers, are expected to attend two different events at one particular time; a room should not be double-booked. A quite comprehensive description of various types of those requirements can be found in [3, 9]. Throughout this paper we will consider the hard con- straints as described in the International Timetabling com- petition [16]. They are: 1. H1: No student clash, i.e. no student is expected to attend two different events at the same time slot; 2. H2: All events should be assigned to a room within the given time slots, and the chosen room for an event should meet the specifications required for that event; 3. H3: A room should not be double-booked. The natural way and perhaps the simplest way of con- structing a feasible solution is to assign each event one by one to its suitable time slot and room until a complete timetable is attained. There are three main concerns in the timetabling construction process. 1. Choosing the order of the events to be assigned; 2. Choosing a time slot for the chosen event; 3. Choosing a suitable room for the chosen event. Many articles relate the feasibility problem to the graph coloring problem [5, 9]. In its basic formulation, a timetable problem can be represented as a graph, where nodes repre- sent events, and there is an edge between two nodes if and only if those two events cannot be assigned to the same time slot due to some constraint. This could happen, for exam- ple, if there is at least one student enrolled in both events or both events are given by the same lecturer. In addition, there could also be an edge between two events, if there is just one suitable room available for them. Thus, two events