Tractability by approximating constraint languages

Abstract

A constraint satisfaction problem instance consists of a collection of variables that need to have values assigned to them. The assignments are limited by constraints that force the values taken by certain collections of variables (the constraint scopes) to satisfy specified properties (the constraint relations). As the general CSP problem is NP-hard there has been significant effort devoted to discovering tractable subproblems of the CSP. The structure of a CSP instance is defined to be the hypergraph formed by the constraint scopes. Restricting the possible structure of the CSP instances has been a successful way of identifying tractable subproblems. The language of a CSP instance is defined to be the set of constraint relations of the instance. Restricting the language allowed for CSP instances has also yielded many interesting tractable subproblems. Almost all known tractable subproblems are either structural or relational. In this paper we construct tractable subproblems of the general CSP that are neither defined by structural nor relational properties. These new tractable classes are related to tractable languages in much the same way that general decompositions (cutset, tree-clustering, etc.) are related to acyclic decompositions. It may well be that our results will begin to make language based tractability of more practical interest. We show that our theory allows us to properly extend the binary maxclosed language based tractable class, which is maximal as a tractable binary constraint language. Our theory also explains the tractability of the constraint representation of the Stable Marriage Problem which has not been amenable to existing explanations of tractability. In fact we provide a uniform explanation for the tractability of the class of maxclosed CSPs and the SMP. There has been much work done on so called renamable HORN theories which are a tractable subproblem of SAT. It has been shown that renamable HORN theories are tractably identifiable and solvable. It has also been shown that finding the largest sub-theory that is renamable HORN is NP-hard. These results also follow immediately from our theory. © Springer-Verlag 2003.