Over-Constrained Systems

Over-constrained systems are unsolvable by traditional methods with respect to the existing contradictory constraints in a problem definition. Solutions of such systems are defined and computed by giving some preferences or weights to the individual constraints and minimizing the violations of constraints using these preferences. The complexity and size of solved problem is often very large and computing of optimal solution requires use of sophisticated methods. Standard methods for solving over-constrained problems apply constraint preferences for definition of optimal solution. There are, however, over-constrained problems with partially or even completely ordered variables and this ordering creates natural preferences for given problem. I developed a new constraint solving environment, where these preferences (variables’ annotations) dependent on variable occurence in constraint are applied (Rudová 1998b). The proposed solution allows to define mapping of variable’s annotations to existing over-constrained systems – constraint hierarchies (Rudová 1998a) and to possibilistic CSPs. With that an algorithm for solving constraints with annotations through mapping to constraint hierarchy was described. This algorithm for acyclic set of inequality constraints was implemented in SICStus Prolog. This part of thesis will conclude with description of some examples demonstrating advantages and scope of annotations and implemented algorithm. The properties of variables’ annotations were still explored on the small or artificial (toy) examples. Currently I try to apply variables’ annotations for solving timetabling problem in our faculty. The working with timetabling problem is a bit more complex than usual, because I would like to create individual timetable for every student – as (s)he registers for a set of courses. Also the number of required courses is small, majority of courses are optional and so the sets of registered courses for each student can be very different. Preferences of variables express how important are given