An Algebraic Theory of Complexity for Discrete Optimization

Abstract

Discrete optimisation problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimised can be expressed as a sum of functions of a restricted form. Here we present a unifying theory of complexity for problems of this kind. We show that the complexity of a finite-domain discrete optimisation problem is determined by certain algebraic properties of the objective function, which we call weighted polymorphisms. We define a Galois connection between sets of rational-valued functions and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised.These results provide a new approach to studying the complexity of discrete optimisation. We use this approach to identify certain maximal tractable subproblems of the general problem, and hence derive a complete classification of complexity for the Boolean case.

Algebraic Properties of Valued Constraint Satisfaction Problem

“…Theorem 9 (Cohen et al [9]). For any finite valued constraint language Γ , we have Imp(wPol(Γ )) = wRelClo(Γ ).…”

“…Following [9] we define a k-ary weighting of a clone C to be a function ω : C (k) → Q such that f ∈C (k) ω(f ) = 0, and if ω(f ) < 0 then f is a projection. The set of operations to which a weighting ω assigns positive weights is called the support of ω and denoted supp(ω).…”

“…We will use the following technical lemma (Lemma 6.5 from [9]). It implies that any weighting that can be expressed as a weighted sum of arbitrary superpositions can also be expressed as a superposition of a weighted sum of proper superpositions.…”

On Some Combinatorial Optimization Problems : Algorithms and Complexity

“…The computational complexity of a CSP is (as we have already mentioned) completely determined by the set of polymorphisms of the constraint language. To allow a similar result for VCSPs several extensions of the concept of polymorphisms has been proposed: multimorphisms [23], fractional polymorphisms [21], and weighted polymorphisms [24]. This has been successful.…”

“…This has been successful. It was proved in [24] that the complexity of VCSP(∆) is completely determined by the weighted polymorphisms of ∆.…”