Quantifiers and approximation

Abstract

We investigate the relationship between logical expressibility of NP optimization problems and their approximation properties. First such attempt was made by Papadimitrou and Yannakakis (1988), who defined the class of NPO problems MAX NP. We show 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 that we consider fit naturally in a new complexity class that we call MAX Π 1. We prove that several natural optimization problems are complete for MAX Π 1 under approximation-preserving reductions. All these complete problems are not approximable unless P = NP. This motivates the definition of subclasses of MAX Π 1 that only contain problems which are presumably eaiser with respect to approximation. In particular, the class that we call RMAX(2) contains approximable problems and problems like MAX CLIQUE that are not known to be nonapproximable. We prove the 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.