Solution equivalent subquadrangle reformulations of constraint satisfaction problems

Abstract

Constraint Satisfaction Problem instances (CSPs) are a natural way to model real life problems such as image processing, scheduling and natural language understanding. In this paper we are concerned with the modeling of problems as CSPs and how this can affect the performance of different solution algorithms. In particular we are interested in modeling in the language of subquadrangles. A Quadrangle is essentially an 'anything-goes' constraint for some Cartesian product of domains. A Subquadrangle [1] is a constraint all of whose projections to proper subsets of the scope are quadrangles. Subquadrangles are a very 'natural' way in which to represent constraints. This is because they do not place any restrictions on proper subsets of their scope, thus reducing the number of required constraint checks. This leads us to believe that a subquadrangle aware solver could be particularly efficient. However, in this paper we shall consider another intriguing use of subquadrangle modeling. Subquadrangles delay the failure of constraint checks until you try and assign a value to the last variable in their scope. This makes them ideal for testing constraint solvers as they are 'backtrack nasty'. In this paper we present two methods by which we might decompose a given instance into a solution equivalent instance whose constraints are all subquadrangles. Our first method converts individual constraints into sets of three subquadrangle constraints. This can then be done for each constraint in a CSP in order to convert the entire problem instance. We are able to demonstrate that this new formulation has a high degree of local consistency related to the arity of the original constraints which induce 'backtrack nasty' behavior in search algorithms. Unfortunately, this with this method one of the subquadrangles is always binary. We also investigate a second method which converts intersecting pairs of constraints. In this method all of the subquadrangles generated are similar in arity to the original constraints. © Springer-Verlag Berlin Heidelberg 2005.