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Volume 3 (2007) Article 3 pp. 45-60
On the Hardness of Satisfiability with Bounded Occurrences in the Polynomial-Time Hierarchy
Received: July 28, 2006
Published: March 28, 2007
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Keywords: satisfiability, polynomial-time hierarchy, expander graphs, superconcentrator graphs
ACM Classification: F.1.3
AMS Classification: 03D15, 68Q17

Abstract: [Plain Text Version]

\newcommand{\myminus}{–} \newcommand{\eps}{\epsilon} \newcommand{\NP}{\mathsf{NP}} \newcommand{\YES}{\mathsf{YES}} \newcommand{\NO}{\mathsf{NO}} \newcommand{\Bsat}{{\mathsf{B}}} \newcommand{\threesat}{\rm{3}\myminus\mathsf{SAT}} \newcommand{\threesatB}{\rm{3}\myminus\mathsf{SAT}\myminus\Bsat} \newcommand{\gapthreesat}{\mathsf{\forall\exists}\myminus{\rm{3}}\myminus\mathsf{SAT}} \newcommand{\gapKthreesatB}{{(\forall\exists)}^r\myminus{\rm{3}}\myminus{\mathsf{SAT}}\myminus\Bsat} \newcommand{\EgapKthreesatB}{\mathsf{\exists(\forall\exists)}^r\myminus{\rm{3}}\myminus\mathsf{SAT}\myminus\Bsat} \newcommand{\gapKDsatB}{{(\forall\exists)}^r\myminus{\rm{\Dsat}}\myminus{\mathsf{SAT}}\myminus\Bsat} \newcommand{\gapthreesatB}{\mathsf{\forall\exists}\myminus{\rm{3}}\myminus\mathsf{SAT}\myminus\Bsat} \newcommand{\gapDsatB}{\mathsf{\forall\exists}\myminus\rm{\Dsat}\myminus\mathsf{SAT}\myminus\Bsat} \newcommand{\gapDEsatB}{\mathsf{\forall\exists}\myminus\rm{\mathsf{E}\Dsat}\myminus\mathsf{SAT}\myminus\Bsat} \newcommand{\gapEthreesatB}{\mathsf{\forall\exists}\rm{\myminus\mathsf{E}3\myminus}\mathsf{SAT}\myminus\Bsat} \newcommand{\gapDsatBX}{\mathsf{\forall\exists}\myminus\rm{\Dsat}\myminus\mathsf{SAT}\myminus\Bsat_\forall} \newcommand{\gapthreesatBX}{\mathsf{\forall\exists}\myminus\rm{3}\myminus\mathsf{SAT}\myminus\Bsat_\forall}

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.