Quantifiers and approximation

Summary form only given. An investigation is made of the relationship between logical expressibility of NP optimization problems and their approximation properties. It is shown that many important optimization problems do not belong to MAX NP and that in fact there are problems in P which are not in MAX NP. The problems considered fit naturally in a new complexity class called MAX II₁. It is proved that several natural optimization problems are complete for MAX II₁ under approximation preserving reductions. All these complete problems are nonapproximable unless P=NP. This motivates the definition of subclasses of MAX II₁ that only contain problems which are presumably easier with respect to approximation. In particular, the class called RMAX(2) contains approximable problems and problems like MAX CLIQUE that are not known to be nonapproximable. It is proven that MAX CLIQUE and several other optimization problems are complete for RMAX(2). All the complete problems in RMAX(2) share the interesting property that they either are nonapproximable or are approximable to any degree of accuracy