Approximation classes for real number optimization problems

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 [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 Blurn, 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 t,o the above mentioned one for combinatorial optimization. We introduce a class NPOℝ of real optimization problems closely related to NPdouble-struck R sign. 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 those classes. © Springer-Verlag Berlin Heidelberg 2006.