Many combinatorial optimization problems, such as the well-known traveling salesman problem (TSP), are defined in terms of minimizing an objective function. Another class of optimization problems are constraint satisfaction problems, which do not have a well-defined objective function. This includes the NP-complete teachers and classes timetabling problem [3, 12, 20]. Briefly, the general problem is stated as follows:
For a certain school withteachers,
classes,
classrooms and lecture halls, and
students, it is required to schedule
teacher-class pairs within a time limit of
time slots producing a legal schedule. A legal schedule needs to be found such that no teacher, class, or student is in more than one place at a time, and no room is expected to accommodate more than one lesson at a time or more students than its capacity.
Figure 1 sketches the approximate relationship between the various entities of the timetabling problem.
Figure 1: The relationship between various entities of the timetabling
problem, where P is professors, C is classes, S is students, R is rooms,
and I is time periods.
The direction of the arrow indicates the direction of the assignment.
Constraint satisfaction problems have have been widely studied in the artificial intelligence community [38, 33, 36]. A solution to a given constraint satisfaction problem is an assignment of values to the variables that is consistent with all of the constraints. Deciding whether or not a given problem instance has a solution is NP-complete in general [36].