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 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 NPO ℝ of real optimization problems closely related to NP ℝ. The class NPO ℝ has four natural subclasses. For each of those we introduce and study real approximation classes APX ℝ and PTAS ℝ 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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Flarup, U., Meer, K. (2006). Approximation Classes for Real Number Optimization Problems. In: Calude, C.S., Dinneen, M.J., Păun, G., Rozenberg, G., Stepney, S. (eds) Unconventional Computation. UC 2006. Lecture Notes in Computer Science, vol 4135. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11839132_8
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DOI: https://doi.org/10.1007/11839132_8
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