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 . 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 . 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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- 3.Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)Google Scholar
- 7.Cucker, F., Rojas, J.M. (eds.): Foundations of Computational Mathematics, Festschrift on occasion of the 70th birthday of Steve Smale. World Scientific, Singapore (2002)Google Scholar
- 8.Li, T.Y.: Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta numerica, vol. 6, pp. 399–436. Cambridge University Press, Cambridge (1997)Google Scholar
- 9.Meer, K.: On some relations between approximation and PCPs over the real numbers. Theory of Computing Systems (to appear)Google Scholar
- 11.Renegar, J., Shub, M., Smale, S. (eds.): Mathematics of Numerical Analysis - Real Number Algorithms. Lecture Notes in Applied Mathematics, vol. 32 (1996)Google Scholar