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

TL;DR

This paper proves that it is computationally hard to distinguish between satisfiable and nearly unsatisfiable instances of a certain logical problem within the polynomial-time hierarchy, even when each variable appears only a bounded number of times.

Contribution

It extends the hardness results for bounded-occurrence SAT problems from NP to higher levels of the polynomial-time hierarchy using superconcentrator graphs.

Findings

Establishes $ ext{Pi}_2$-hardness for bounded-occurrence $ ext{gap3SAT}$ instances.

Generalizes the hardness result to all levels of the polynomial-time hierarchy.

Provides a new tool for deriving inapproximability results in complexity theory.

Abstract

$ \newcommand{\eps}{\epsilon} \newcommand{\NP}{\mathsf{NP}} \newcommand{\YES}{\mathsf{YES}} \newcommand{\NO}{\mathsf{NO}} \newcommand{\myminus}{\text{-}}\newcommand{\Bsat}{{\mathsf{B}}} \newcommand{\threesat}{\rm{3}\myminus\mathsf{SAT}} \newcommand{\gapthreesat}{\mathsf{\forall\exists}\myminus{\rm{3}}\myminus\mathsf{SAT}} $In 1991, Papadimitriou and Yannakakis gave a reduction implying the $\NP$-hardness of approximating the problem $\threesat$ 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 $\gapthreesat$ in which every variable occurs at most $\Bsat$ times (for some absolute constant $\Bsat$), it is $\Pi_2$-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 $1-\eps$ fraction of the clauses. We also generalize this result to any level of the polynomial-time hierarchy.