On the Hardness of Satisfiability with Bounded Occurrences in the Polynomial-Time Hierarchy

In 1991, Papadimitriou and Yannakakis gave a reduction implying the NPhardness 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 ∗Supported by an Alon Fellowship, by the Binational Science Foundation, by the Israel Science Foundation, and by the EU Integrated Project QAP. †Supported by the Binational Science Foundation, by the Israel Science Foundation, and by the EU Integrated Project QAP. ACM Classification: F.1.3 AMS Classification: 03D15, 68Q17