Encodings of Non-Binary Constraint Satisfaction Problems

We perform a detailed theoretical and empirical comparison of the dual and hidden variable encodings of non-binary constraint satisfaction problems. We identify a simple relationship between the two encodings by showing how we can translate between the two by composing or decomposing relations. This translation suggests that we will tend to achieve more pruning in the dual than in the hidden variable encoding. We prove that achieving arc-consistency on the dual encoding is strictly stronger than achieving arc-consistency on the hidden variable, and this itself is equivalent to achieving generalized arc-consistency on the original (non-binary) problem. We also prove that, as a consequence of the unusual topology of the constraint graph in the hidden variable encoding, inverse consistencies like neighborhood inverse consistency and path inverse consistency collapse down onto arc-consistency. Finally, we propose the “double encoding”, which combines together both the dual and the hidden variable encodings.