Bayes networks for estimating the number of solutions of constraint networks

The problem of counting the number of solutions to a constraint network (CN) (also called constraint satisfaction problems, CSPs) is rephrased in terms of probability updating in Bayes networks. Approximating the probabilities in Bayes networks is a problem which has been studied for a while, and may well provide a good approximation to counting the number of solutions. We use a simple approximation based on independence, and show that it is correct for tree-structured CNs. For other CNs, it is less optimistic than a spanning-tree approximation suggested in prior work. Experiments show that the Bayes nets estimation is more accurate on the average, compared to competing methods (the spanning-tree, as well as a scheme based on a product of all compatible pairs of values). We present empirical evidence that our approximation may also be a useful (value ordering) search heuristic for finding a single solution to a constraint satisfaction problem.