Hardness of approximating /spl Sigma//sub 2//sup p/ minimization problems

We show that a number of natural optimization problems in the second level of the Polynomial Hierarchy are /spl Sigma//sub 2//sup p/-hard to approximate to within n/sup /spl epsiv// factors, for specific /spl epsiv/>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/sup 1-/spl epsiv// 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/sup /spl epsiv// factor using the same technique.