Approximation Classes for Real Number Optimization Problems

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 [1]. 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 [4]. 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.