An Algebraic Theory of Complexity for Discrete Optimization

Discrete optimization problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimized 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...

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Veröffentlicht in:SIAM journal on computing Jg. 42; H. 5; S. 1915 - 1939
Hauptverfasser: Cohen, David A., Cooper, Martin C., Creed, Páidí, Jeavons, Peter G., Živný, Stanislav
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
Schlagworte:
Algebra
Applied mathematics
Artificial Intelligence
Boolean
Cloning
Computation and Language
Computer Science
Integer programming
International conferences
Language
Logic in Computer Science
Machine Learning
Optimization
Polymorphism
Scholarships & fellowships
Values
Weighted Polymorphisms
Galois Connection
Valued Constraint Satisfaction Problems
Constraint Optimization
Discrete Optimization
Weighted Clones
ISSN:0097-5397, 1095-7111
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Zusammenfassung:Discrete optimization problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimized 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 optimization 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 characterized. These results provide a new approach to studying the complexity of discrete optimization. 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. [PUBLICATION ABSTRACT]
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ISSN:0097-5397
1095-7111
DOI:10.1137/130906398