On the Parallel Complexity of Some Constraint Satisfaction Problems

Constraint satisfaction networks have been shown to be a very useful tool for knowledge representation in Artificial Intelligence applications. These networks often utilize local constraint propagation techniques to achieve global consistency (consistent labelling in vision). Such methods have been used extensively in the context of image understanding and interpretation, as well as planning, natural language analysis and commonsense reasoning. In this paper we study the parallel complexity of discrete relaxation, one of the most commonly used constraint satisfaction techniques. Since the constraint propagation procedures such as discrete relaxation appear to operate locally, it has been previously believed that the relaxation approach for achieving global consistency has a natural parallel solution. Our analysis suggests that a parallel solution is unlikely to improve by much the known sequential solutions. Specifically, we prove that the problem solved by discrete relaxation is logspace complete for P (the class of polynomial time deterministic sequential algorithms). Intuitively, this implies that discrete relaxation is inherently sequential and it is unlikely that we can solve the polynomial time version of the consistent labelling problem in logarithmic time by using only a polynomial number of processors. Some practical implications of our result are discussed.