In 1991, Papadimitriou and Yannakakis gave a reduction implying the NP-hardness of approximating the problem 3-SAT with bounded occurrences. Their reduction is based on expander graphs. We present an analogue of this result for the second level of the polynomial-time hierarchy based on superconcentrator graphs. This resolves an open question of Ko and Lin (1995) and should be useful in deriving inapproximability results in the polynomial-time hierarchy.

More precisely, we show that given an instance of ∀∃-3-SAT in which every variable occurs at most B times (for some absolute constant B), it is Π₂-hard to distinguish between the following two cases: YES instances, in which for any assignment to the universal variables there exists an assignment to the existential variables that satisfies all  the clauses, and NO instances in which there exists an assignment to the universal variables such that any assignment to the existential variables satisfies at most a (1-ε) fraction of the clauses. We also generalize this result to any level of the polynomial-time hierarchy.