Probabilistic Arc consistency

Abstract

The two most popular algorithms for solving Constraint Satisfaction Problems are Forward Checking (FC) [1] and Maintaining Arc Consistency (MAC) [2]. MAC maintains full arc consistency while FC maintains a limited form of arc consistency during search. There is no single champion algorithm: MAC is more efficient on sparse problems which are tightly constrained but FC has an increasing advantage as problems become dense and constraints loose. Ideally a good search algorithm should find the right balance - for any problem - between visiting fewer nodes in the search tree and reducing the work that is required to establish local consistency. In order to do so, we maintain probabilistic arc consistency during search. The idea is to assume that a support exists and skip the process of seeking a support if the probability of having some support for a value is at least equal to some, carefully chosen, stipulated threshold. Arc consistency involves revisions of domains, which require support checks to remove unsupported values. In many revisions, some or all values find some support. If we can predict the existence of a support then a considerable amount of work can be saved. In order to do so, we propose the notions of a Probabilistic Support Condition (PSC) and Probabilistic Revision Condition (PRC). If PSC holds then the probability of having some support for a value is at least equal to the threshold and the process of seeking a support is skipped. If PRC holds then for each value the probability of having some support is at least equal to the threshold and the corresponding revision is skipped. For hard dense problems constraint are generally loose, and on average each value has several supports, in which case the probability of having some support remains high for a while with respect to the number of values removed. This enables PSC and PRC to save checks. For hard sparse problems constraints are generally tight, and on average each value has only a few supports, in which case the probability of having some support drops rapidly with respect to the number of values removed. This forces both PSC and PRC to fail quickly, which in turn forces the algorithm to behave like MAC. Unlike MAC and FC where the strength of constraint propagation is static, maintaining probabilistic arc consistency allows to adjust the strength of constraint propagation dynamically during search and performs well on both dense and sparse problems. © Springer-Verlag Berlin Heidelberg 2005.