Approximation properties of NP minimization classes

The authors introduce a novel approach to the logical definability of NP optimization problems by focusing on the expressibility of feasible solutions. They show that in this framework first-order sentences capture exactly all polynomially bounded optimization problems. They also show that, assuming P≠NP, it is an undecidable problem to determine whether a given first-order sentence defines an approximable optimization problem. They then isolate a syntactically defined class of NP minimization problems that contains the min set cover problem and has the property that every problem in it has a logarithmic approximation algorithm. They conclude by giving a machine-independent characterization of the NP=co-NP problem in terms of logical expressibility of the max clique problem