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
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.