Directed Arc Consistency Preprocessing as a Strategy for Maximal Constraint Satisfaction

1 ABSTRACT A constraint satisfaction problem (CSP) may be overcon-strained and not admit a complete solution. Optimal solutions to such partial constraint satisfaction problems (PCSPs), in which a maximum number of constraints are satissed, can be found using branch and bound variants of CSP algorithms. Earlier work has shown how information gained through local consistency checking during preprocessing can be used to enhance search through value ordering heuristics and local lower bound calculations that involve only neighboring variables. The present work describes a family of strategies based on directed arc consistency testing during pre-processing. With this approach inconsistency counts associated with each value (the number of domains that ooer no support for that value) are obtained that are non-redundant, since they are unidirectional. They can, therefore, be used to obtain global lower bounds that involve the entire set of variables. By computing directed arc consistency in each direction, full arc-inconsistency counts can also be obtained, thus retaining the beneets of full arc consistency checking, while improving lower bound calculations. Retrospective and prospective algorithms that incorporate the results of directed arc consistency checking are described. Tests with random problems show improvements, sometimes marked, over the best branch and bound PCSP algorithms heretofore described. 1.1 Introduction Constraint satisfaction problems (CSPs) involve assigning values to variables which satisy a set of constraints. A partial constraint satisfaction problem (PCSP) is one that is 'overconstrained', so there is no assignment that can satisfy all its constraints. In this case it may still be useful to have an assignment of values to variables that satisses as many constraints as