Hardness of approximating Σ2p minimization problems

We show that a number of natural optimization problems in the second level of the Polynomial Hierarchy are Σ₂^p -hard to approximate to within n^ε factors, for specific ε>0. The main technical tool is the use of explicit dispersers to achieve strong, direct inapproximability results. The problems we consider include Succinct Set Cover, Minimum Equivalent DNF, and other problems relating to DNF minimization. Under a slightly stronger complexity assumption, our method gives optimal n^1-ε inapproximability results for some of these problems. We also prove inapproximability of a variant of an NP optimization problem, Monotone Minimum Satisfying Assignment, to within an n^ε factor using the same technique