This paper investigates the subclasses that arise from restricting the possible constraint types, and shows that any set of constraints that does not give rise to an NP-complete class of problems must satisfy a certain type of algebraic closure condition.Expand

A simple algebraic property is described which characterises all possible constraint types for which strong k-consistency is sufficient to ensure global consistency, for each k > 2.Expand

It is shown that many tractable sets of soft constraints, both established and novel, can be characterised by the presence of particular multimorphisms and the notion of multimorphism is used to give a complete classification of complexity for the Boolean case which extends several earlier classification results for particular special cases.Expand

It is proved that a constraint satisfaction problem may be decomposed into a number of subproblems precisely when the corresponding hypergraph satisfies a simple condition.Expand

A generic form of structural decomposition for the constraint satisfaction problem, which is a guarded decomposition, and it is shown that many existing decomposition methods can be characterised in terms of finding guarded decompositions satisfying certain specified additional conditions.Expand

The many different definitions of symmetry for constraint satisfaction problems (CSPs) are reviewed, and it is shown that a symmetry can be defined in two fundamentally different ways: as an operation preserving the solutions of a CSP instance, or else as anoperation preserving the constraints.Expand

A new way of identifying tractable subproblems of the Constraint Satisfaction Problem is defined and the notion of a ''lifted constraint instance'' which is a powerful tool to study this question is described.Expand

A set of constraints is identified which gives rise to a class of tractable problems and given polynomial time algorithms for solving such problems, and it is proved that the class of problems generated by any set of constraint not contained in this restricted set is NP-complete.Expand

This chapter investigates how the complexity of solving constraint problems varies with the types of constraints which are allowed, and considers some alternative approaches, including a constructive approach that builds new tractable constraint languages by combining simpler languages.Expand

This paper demonstrates that this general form of submodular function minimization problem is just one example of a much larger class of tractable discrete optimization problems defined by valued constraints, which includes a tournament pair multimorphism.Expand