An Algebraic Theory of Complexity for Valued Constraints: Establishing a Galois Connection

The Complexity of Valued Constraint Satisfaction Problems

“…2.4 and 2.5 together, Sect. 2.6 presents a Galois connection between sets of valued constraints and sets of algebraic closure operations [68].…”

“…Using the algebraic approach to characterising the expressive power of valued constraints developed by Cohen et al in [65,68] and discussed in Chap. 2, we are able to give a negative answer to Problem 5.1: we show that there exist submodular polynomials of arity 4 that cannot be expressed by binary submodular polynomials.…”

“…2.3. A characterisation of this Galois connection for finite sets D is given by the following two theorems [68].…”

“…Chapter 2, based on [63,68,274], presents an algebraic theory for the expressive power of languages. Chapter 3, based on [70] (preliminary version [69]) and [272], investigates the expressive power of fixed-arity languages.…”

“…This connection has been successfully used in the last decade for complexity analysis of crisp constraint languages [42,46]. This chapter presents a recently developed algebraic theory for the complexity of valued constraint languages [63,68], which builds on [65,274].…”