Approximation Properties of NP Minimization Classes

In this paper we study NP optimization problems from the perspective of descriptive complexity theory. We introduce a new approach to the logical definability of NP optimization problems by focusing on the ways in which the feasible solutions of the problems can be described using first-order formulae. We delineate the expressive power of this new framework and determine the relations between the various classes of optimization problems arising in this context. We also show that, assuming that P ? NP, it is an undecidable problem to tell if a given first-order formula defines an approximable NP optimization problem. After this, we investigate the impact that syntactic or structural properties of formulae have on the approximation properties of optimization problems defined by the formulae. Our main finding is that the approximation properties of many natural NP minimization problems are due to the fact that they are definable in the new framework using positive first-order formulae. In particular, we isolate a syntactically defined class of NP minimization problems that contains the MIN SET COVER problem and has the property that all its members have a polynomial-time O(log n)-approximation algorithm. Finally, we pursue results in a different direction and obtain a machine-independent characterization of the NP = ? co-NP problem in terms of logical definability of the MAX CLIOUE problem.