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An Algebraic Theory of Complexity for Discrete Optimisation

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.