Fundamental Properties of Neighbourhood Substitution in Constraint Satisfaction Problems

In combinatorial problems it is often worthwhile simplifying the problem, using operations such as consistency, before embarking on an exhaustive search for solutions. Neighbourhood substitution is such a simplification operation. Whenever a value x for a variable is such that it can be replaced in all constraints by another value y, then x is eliminated. This paper shows that neighbourhood substitutions are important whether the aim is to find one or all solutions. It is proved that the result of a convergent sequence of neighbourhood substitutions is invariant modulo isomorphism. An efficient algorithm is given to find such a sequence. It is also shown that to combine consistency (of any order) and neighbourhood substitution, we only need to establish consistency once. 1. Neighbourhood substitution In a CSP (constraint satisfaction problem) on y1 variables, constraints are given in the form of relations C(P,), . . . , C(P,) on subsets P,, . . . , P, of the n variables. C(P,) is the set of legal labellings for the set of variables P,. It may be given as an explicit list of labellings or in implicit closed form. The set of n-tuples satisfying all the constraints is simply the join of the constraints C(P,), .. . , C(P,). The domain of variable i is denoted by Ai. Many problems, such as schooltimetabling, scheduling, line-drawing labelling, sketch-map interpretation and circuit design can be expressed in a natural way as CSPs. For example, in the graph colouring problem, in which the aim is to assign a colour to each node of a given graph G so that no two adjacent nodes are assigned the same colour, there is a binary constraint * E-mail: [email protected]. 0004-3702/97/$17.00