Logical Definability of NP Optimization Problems

We investigate here NP optimization problems from a logical deenability standpoint. We show that the class of optimization problems whose optimum is deenable using rst-order formulae coincides with the class of polynomially bounded NP optimization problems on nite structures. After this, we analyze the relative expressive power of various classes of optimization problems that arise in this framework. Some of our results show that logical deenability has diierent implications for NP maximization problems than it has for NP minimization problems, in terms of both expressive power and approximation properties.