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Logical Definability of NP Optimization Problems

We investigate here NP optimization problems from a logical definability standpoint. We show that the class of optimization problems whose optimum is definable using first-order formulae coincides with the class of polynomially bounded NP optimization problems on finite 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 definiability has different implications for NP maximization problems than it has for NP minimization problems, in terms of both expressive power and approximation properties.