Improving Cost Calculations for Global Constraints in Local Search

Abstract

Local search for constraint satisfaction is usually performed by local minimization of a cost function. Traditionally, the cost function used is simply the number of violated constraints for a given assignment. When introducing global constraints in local search, use of this cost function will give the effect that practically no global constraints will be satisfied. The reason for this is that global constraints in general are much harder to satisfy than other constraints, and are as a consequence of this also hard to satisfy within the limited neighborhood of the current assignment. Another reason is that the value of satisfying a global constraint is as low as for any other constraint. Specific cost functions for global constraints have been proposed to amend this. The total cost of an assignment using this scheme is the sum of the costs of the constraints present. Instead of using a specific cost function for each global constraint, we propose to use a generic cost function, based on a formal definition of the specific global constraint used. We use a framework based on set reasoning regarding certain basic constraints to formally define global constraints. From this definition, we then calculate a fair virtual cost, based on the number of basic constraints that are violated. In addition to using the virtual cost to guide local search, we can use this cost function for labeling heuristics, such as the most constrained first strategy found in many traditional constraint programming systems