Soft Constraints: Complexity and Multimorphisms

Over the past few years there has been considerable progress in methods to systematically analyse the complexity of classical (crisp) constraint satisfaction problems with specified constraint types. One very powerful theoretical development in this area links the complexity of a set of classical constraints to a corresponding set of algebraic operations, known as polymorphisms. In this paper we begin a systematic investigation of the complexity of combinatorial optimisation problems expressed using various forms of soft constraints. We extend the notion of a polymorphism by introducing a more general algebraic operation, which we call a multimorphism. We show that a number of maximal tractable sets of soft constraints, both established and novel, can be characterised by the presence of particular multimorphisms.

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