Non-Binary Constraints

Since the origins of the constraint satisfaction paradigm, its restriction to binary constraints has concentrated a signiicant part of the work. This is understandable because new ideas/techniques are usually much simpler to present/ elaborate by rst restricting them to the binary case. (See for example the arc consistency algorithms, such as AC-3 or AC-4, which have been presented rst in their binary version 10, 12], before being extended to non-binary constraints 11, 13].) But this inclination has highly increased in the early nineties. Authors indeed justiied this restriction by the fact that any non-binary constraint network can polyniomally be converted into an equivalent binary one 6, 8, 5, 19]. And, in most cases, they never extended their work to non-binary constraints. Up to now, constraint reasoning has generated robust formal deenitions (local consistencies, etc.), original resolution methods ((ltering, look-back schemes, decomposition techniques, heuristics, etc.), and theoretical results on tractable classes. They were proved useful on many academic problems, and are probably at the origin of the success of this area. But now, constraint reasoning is going towards its maturity, and should then be able to propose eecient resolution techniques for real-world problems, for which the binary conversion is sometimes/often impracticable. This observation, stressed by van Beek in its CP'96 invocation, has started having its eeect since papers especially devoted to non-binary constraints or to their binary conversion begin to appear in this end of decade 4, 1, 22, 2]. Dealing with non-binary constraints for real-world problems raises questions that do not appear as crucial on binary constraints. Applying ltering techniques during search is central in the eeciency of constraint satisfaction algorithms. But applying ltering on non-binary constraints is much more expensive than on binary ones. The weaknesses of the algorithms appear more accurately. Does this mean that we must circumscribe ltering to its weakest form, as in the classical extension to non-binary constraints of the forward checking algorithm? The answer is no. This algorithm is much more subject to thrashing than on binary networks 1]. Local consistencies, which are one of the strengths of constraint reasoning, must also be applied in the non-binary case. But they have perhaps to be seen in a new way.