Domain permutation reduction for constraint satisfaction problems

Abstract

This paper is concerned with the Constraint Satisfaction Problem (CSP). It is well-known that the general CSP is NP-hard. However, there has been significant success in discovering subproblems which are tractable (polynomial time solvable). One of the most effective ways to obtain a tractable subproblem has been to force all of the constraint relations to lie in some tractable language. In this paper we define a new way of identifying tractable subproblems of the CSP. Let P be an arbitrary CSP instance and Γ be any tractable language. Suppose there exists, for each variable of P, a permutation of the domain such the resultant permuted constraint relations of P all lie in Γ. The domain permuted instance is then an instance of a tractable class and can be solved by the polynomial time algorithm for Γ. Solutions to P can be obtained by inverting the domain permutations. The question, for a given class of instances and language Γ, whether such a set of domain permutations can be found efficiently is the key to this method's tractability. One of the important contributions made in this paper is the notion of a “lifted constraint instance” which is a powerful tool to study this question. • We consider the open problem of discovering domain permutations which make instances max-closed. We show that, for bounded arity instances over a Boolean domain this problem is tractable, while for domain size three it is intractable even for binary instances. • We give a simple proof verifying the tractability of discovering domain permutations which make instances row convex. We refute a published result by giving a simple proof of the intractability of discovering domain permutations which make instances, even with domain size four, connected row convex. • We demonstrate that triangulated and stable marriage instances are reducible, via domain permutations, to max-closed instances. This provides a simple explanation for arc consistency deciding these instances. • We verify with a simple direct proof the tractability of identification of renamable Horn instances, and the intractability of finding the largest renamable Horn subset of clauses of an instance of SAT. • We describe natural tractable classes which properly extend the maximal relational classes arising from tractable constraint languages. We believe that domain permutation reductions have a significant chance of being useful in practical applications.