Optimization, Approximation, and Complexity Classes

W e 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. W e 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 MAXSNP under a kind of careful transformation (called L-redudon) that preserves approximability. It follows that such a complete problem has a polynomial-t ime approximation scheme iff the whole class does. These results may help explain the lack of progress on the approximabil ity of a host of optimization problems.