Arc Consistency in the Dual Encoding of Non-Binary CSPs

A non-binary Constraint Satisfaction Problem (CSP) can be solved by converting the problem into an equivalent binary one and applying well-established binary CSP techniques. An alternative way is to use extended versions of binary techniques directly in the non-binary problem. There are two well-known methods in the literature for translating a non-binary CSP to an equivalent binary one; the hidden variable encoding and the dual encoding. It has been shown that arc consistency can be applied in the hidden variable encoding of a non-binary CSP with the same worst-case time complexity as generalized arc consistency in the non-binary representation. However, arc consistency in the dual encoding is so far considered prohibitively expensive to apply. In this paper we describe an arc consistency algorithm for the dual encoding with O(ed) worst-case complexity. This gives an O(d/e) saving compared to a generic algorithm and is close to the complexity of generalized arc consistency in the non-binary representation. Experimental results show that the new algorithm can be orders of magnitude better than an optimal generic arc consistency algorithm. Also, due to the stronger filtering achieved in the dual encoding compared to the non-binary representation, utilization of the new algorithm can make the dual encoding a competitive option for certain classes of non-binary constraints.