Solving functional constraints by variable substitution

Abstract

Functional constraints and bi-functional constraints are an important constraint class in Constraint Programming (CP) systems, in particular for Constraint Logic Programming (CLP) systems. CP systems with finite domain constraints usually employ CSP-based solvers which use local consistency, for example, arc consistency. We introduce a new approach which is based instead on variable substitution. We obtain efficient algorithms for reducing systems involving functional and bi-functional constraints together with other non-functional constraints. It also solves globally any CSP where there exists a variable such that any other variable is reachable from it through a sequence of functional constraints. Our experiments on random problems show that variable elimination can significantly improve the efficiency of solving problems with functional constraints.

Variable Elimination in Binary CSPs

“…Fourier's algorithm for variable elimination applied to a system of binary linear inequalities (Koubarakis, 2006;Schrijver, 1999) can be viewed as just one example of this general rule, since binary linear inequalities are all closed under the majority polymorphism median. Another interesting case is when there is a functional constraint of the form x i = f (x j ) (where f is a function) for some other variable x j : the relation R −x i is then equivalent to the join of its projections onto the pairs of variables (x j , x k ) (k = i, j) (Zhang & Yap, 2011).…”