Arc consistency is not really a good term to use, since it was originally defined for 2-ary constraints only, but is sometimes used for n-ary constraints. That

Arc consistency is not really a good term to use, since it was originally defined for 2-ary constraints only, but is sometimes used for n-ary constraints. That is why I use the term domain consistency consistently :-) Given a propagator for circuit that reaches domain consistency (note, for the whole n-ary constraint), that is equivalent to a checker of there is a Hamiltonian path and that will most likely never be a polynomial algorithm (unless P=NP). I can’t say that I’m interested in any particular property of the decomposition of the constraint into 2-ary parts, it is not that interesting of a structure. The cool filterings that circuit propagators do are based on global analysis of the whole graph, not on any local property. For example by finding ordering a of strongly connected components in the current graph. Almost all propagators need to be reasonably fast to be usable, that is sort of a given in this area. An exponential propagator is almost always a bad idea. For all_different there is a O(n^2.5) algorithm for domain consistency which is really cool and important. In practice, the simple value propagation (remove assigned values from the other variables) is often better for finding solutions quickly.