Approximation classes for real number optimization problems
Klaus Meer and Uffe Flarup
In: 5th International Conference on Unconventional Computation,, 4-8 Sep 2006, York, UK.

Abstract

A fundamental research area in relation with analyzing the complexity of optimization problems are approximation algorithms. For combinatorial optimization a vast theory of approximation algorithms has been developed, see \cite{Ausiello}. Many natural optimization problems involve real numbers and thus an uncountable search space of feasible solutions. A uniform complexity theory for real number decision problems was introduced by Blum, Shub, and Smale \cite{BSS}. However, approximation algorithms were not yet formally studied in their model. In this paper we develop a structural theory of optimization problems and approximation algorithms for the BSS model similar to the above mentioned one for combinatorial optimization. We introduce a class $\npor$ of real optimization problems closely related to $\npr.$ The class $\npor$ has four natural subclasses. For each of those we introduce and study real approximation classes $\apxr$ and $\ptasr$ together with reducibility and completeness notions. As main results we establish the existence of natural complete problems for all these classes.

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