Approximable Minimization Problems and Optimal Solutions on Random Inputs

In this paper we extend recent work about logical criteria for approximation properties of optimization problems. We focus on the relationship between logical expressibility and expected asymptotic growth of optimal solutions on random inputs. This further develops a probabilistic approach due to Behrendt, Compton and Grr adel showing that expected optimal solutions for any problem in the class Max 1 grows essentially like a polynomial. We show that there is a similar result for Min F + 1, a syntactic class of minimization problems which provides a logical criterion for approximability. As a consequence, we show that some important problems do not belong to Min F + 1.