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Bayes networks for estimating the number of solutions of constraint networks

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Abstract

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.

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Authors
  1. Amnon Meisels
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  2. Solomon Eyal Shimony
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  3. Gadi Solotorevsky
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Meisels, A., Shimony, S.E. & Solotorevsky, G. Bayes networks for estimating the number of solutions of constraint networks. Annals of Mathematics and Artificial Intelligence 28, 169–186 (2000). https://doi.org/10.1023/A:1018956222900

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