A Unifying Framework for Structural Properties of CSPs: Definitions, Complexity, Tractability

Abstract

Literature on Constraint Satisfaction exhibits the definition of several "structural" properties that can be possessed by CSPs, like (in)consistency, substitutability or interchangeability. Current tools for constraint solving typically detect such properties efficiently by means of incomplete yet effective algorithms, and use them to reduce the search space and boost search. In this paper, we provide a unifying framework encompassing most of the properties known so far, both in CSP and other fields' literature, and shed light on the semantical relationships among them. This gives a unified and comprehensive view of the topic, allows new, unknown, properties to emerge, and clarifies the computational complexity of the various detection problems. In particular, among the others, two new concepts, fixability and removability emerge, that come out to be the ideal characterisations of values that may be safely assigned or removed from a variable's domain, while preserving problem satisfiability. These two notions subsume a large number of known properties, including inconsistency, substitutability and others. Because of the computational intractability of all the property-detection problems, by following the CSP approach we then determine a number of relaxations which provide sufficient conditions for their tractability. In particular, we exploit forms of language restrictions and local reasoning.

Replaceability and the Substitutability Hierarchy for Constraint Satisfaction Problems

“…a total order, or more simply a hierarchy) in which the substitutability property pertains to progressively more values. In addition, this work extends the characterization of the replaceability property highlighted by [1], and shows that it has certain 'maximal' features within the series with regard to computability. This confirms the contention of these authors that this is, indeed, a property of special significance.…”

“…The earliest is the work by Jeavons et al, who defined what they called "generalized substitutability" [5]. Later Bordeaux et al defined a similar but simpler concept that they called "removability" [1]. In both cases, the basic idea is that, given value v that can appear in some solution, then for any solution that v appears in there is some other value that can be substituted for v, giving another valid solution.…”

“…Then [5] would allow the set b, c to substitute for a. While this situation is not ruled out by our formulation, it does involve a value that is 'irrelevant' to the definition of replaceability given above (and by [1]). This leads to the following observation.…”

“…The first paper on this subject defined two basic properties involving either equivalence or subsumption of single values; these were called interchangeability and substitutability, respectively [2]. Since then a large collection of related properties have been proposed, and subsequent to this various dominance and incomparability relations have been established; these are summarized in [7] Some years ago Bordeaux, Cadoli, and Mancini proposed a general framework for these and other properties [1]. They identified what they considered a key concept that they called "removeability" and that we will refer to as replaceability, which is a generalized form of substitutability.…”

“…In the present work, which can be seen as an adjunct to the work of both [1] and [3], we focus on the basic property of substitutability and its generalizations, all of which involve relaxing the conditions under which we may remove a given value (or set of values). The reason for considering this special subset of properties, which forms a relatively small subgraph in the network of properties discussed by [7], is that this set of properties has a well-formed set of interrelations so that they can be arranged in a single series (i.e.…”