Quantifiers and approximation

We investigate tile relationship between logical express-ibility of NP optimization problems and their approximation properties. First sucll attempt was made by Papadimitriou and Yannakakis, who defined the class of NPO problems MAX NP. We show that many impor-taut optimization problems do not belong to MAX NP and that in fact there are problems in P which are not ill lk'IAX NP. The problems that we consider fit naturally in a new complexity class that we call MAX Ill. We prove that several natural optimization problems are complete for MAX H1 under approxima.tion preserving reductions. All these complete problems are non approximable 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 that we call RMAX(2), contains approximable problems and prob-]elllS like MAX CLIQUE that are not known to be non-approximable. We prove that MAX CLIQUE and several other optimization problems are complete for RMAX(2). Permission to copy without fee all or part of this matertial is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. All the complete problems in RMAX(2) share the interesting property that they either are non-approximable or are a.pproximable to ally degree of ac-cura.cy.