Fundamental properties of neighbourhood substitution in constraint satisfaction problems

Reviewer: Matthew Mark Huntbach

A constraint satisfaction problem (CSP) consists of a set of variables, each of which has a set of possible values, and a set of rules over subsets of the variables which constrain the values they may have. The task is to find variable assignments that satisfy the rules. Graph coloring, in which the variables are the nodes and the rules state that no connected nodes may have the same value, is one of many problems expressible as a CSP. In general, CSPs are NP-complete, but in practice, various methods can reduce the search space of possible solutions to a manageable size. This paper proves some results regarding one such method, neighborhood substitution [1], in which a value may be removed from the set for a variable because it is, in a sense, just a variant of another. The author shows that the order in which neighborhood substitutions are applied is irrelevant, but k -consistency operations (another search space reduction method) should be applied first. An algorithm to find all solutions to a CSP taking advantage of neighborhood substitutions is given. Inevitably for a paper on proof of algorithms, this work requires a fair amount of mathematical competence to follow. The references are well chosen, and the casual reader should prepare for the paper by reading some of them first, as they are the foundation on which the paper builds.