On Consistency and Width Notions for Constraint Programs with Algebraic Constraints

• Authors: Klaus Meer
• Venue: FLOPS '02 Proceedings of the 6th International Symposium on Functional and Logic Programming
• Year: 2002

Width notions have been studied in the framework of constraint satisfaction problems in order to exhibit subclasses of problems which allow backtrack-free solution algorithms. Among the most prominent such notions is the tree-width of the constraint graph studied, for example, by Freuder [6, 7]. However, Freuder's results heavily rely on constraint programs over finite domains, where each constraint is given as a list of admissible tuples and therefore fails, for example, if continuous domains are considered. Faltings [5] introduced an arc consistency notion for constraints over continuous domains that are given in a more complicated form using formulas c(x, y) 驴 0 for continuously differentiable functions c. He then showed for such binary constraints how arc consistency can be established and guarantees solvability of tree-structured problems.In this paper we want to study a generalization of Freuder's and Faltings' notions to problems with algebraic constraints. We show that an analog notion of k-consistency guarantees backtrack-free solution algorithms for tree-structured problems, but argue that already for binary constraints and a tree as structure of the constraint graph there arise unavoidable complexity problems in achieving k-consistency. We then propose a new width notion based on [10] which in certain situations even allows to include global constraints without yielding a complexity explosion - something not true within the above mentioned setting.