We define a natural variant of NP, MAX NP, and also a subclass called MAX SNP. These are classes of optimization problems, and in fact contain several natural, well-studied ones. We show that problems in these classes can be approximated with some bounded error. Furthermore, we show that a number of common optimization problems are complete for MAX SNP under a kind of careful transformation (called L-reduction) that preserves approximability. It follows that such a complete problem has a polynomial-time approximation scheme iff the whole class does. These results may help explain the lack of progress on the approximability of a host of optimization problems. © 1991.
Papadimitriou, C. H., & Yannakakis, M. (1991). Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 43(3), 425–440. https://doi.org/10.1016/0022-0000(91)90023-X