An Algebraic Theory of Complexity for Discrete Optimisation

David A. Cohen; Martin C. Cooper; Paidi Creed; Peter G. Jeavons; Stanislav Zivny

doi:10.1137/130906398

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arXiv · 1207.6692

An Algebraic Theory of Complexity for Discrete Optimisation

David A. Cohen·Martin C. Cooper·Paidi Creed·Peter G. Jeavons·Stanislav Zivny

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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.

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David A. Cohen, Martin C. Cooper, Paidi Creed, Peter G. Jeavons, Stanislav Zivny. 2012-07-28. An Algebraic Theory of Complexity for Discrete Optimisation. https://doi.org/10.1137/130906398

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