An Incremental and Non-binary CSP Solver: The Hyperpolyhedron Search Algorithm

Abstract

Nowadays, many real problems can be efficiently modelled as Constraint Satisfaction Problems (CSP’s). Most of these problems can be naturally modeled using non-binary constraints. However, researchers traditionally had focused on binary constraints, due to the simplicity of dealing with binary constraints and the fact that any non-binary CSP can be transformed into an equivalent binary one. Moreover, this transformation may not be practical in some problems, because the binarized CSP produces a significant increase in the problems size and the translation process generates new variables, which may have very large domains. Thus, it becomes necessary to manage problems with non-binary constraints directly. In this work, we propose an algorithm called “Hyperpolyhedron Search Algorithm (HSA)” that solves non-binary constraint satisfaction problems in a natural way as an incremental and non-binary CSP solver. HSA is a constraint propagation algorithm that carries out the search through a hyperpolyhedron that maintains in its vertices those solutions that satisfy all non-binary constraints. In HSA, the handling of the non-binary constraints (linear inequations) can be seen as a global hyperpolyhedron constraint. Initially, the hyperpolyhedron is created by the Cartesian Product of the domain bounds of variables. For each constraint, HSA checks the consistency, updating the hyperpolyhedron (if the constraint is consistent), by means of LP techniques. This constraint is a hyperplane that is intersected to obtain the new hyperpolyhedron vertices. The resulting hyperpolyhedron is a convex set of solutions to the CSP.