Completeness in approximation classes
Information and Computation
On the computational complexity of approximating solutions for real algebraic formulae
SIAM Journal on Computing
Complexity and real computation
Complexity and real computation
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Some Relations between Approximation Problems and PCPs over the Real Numbers
Theory of Computing Systems
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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.