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TR06-160 | 17th December 2006 00:00

Closures and dichotomies for quantified constraints



Quantified constraint satisfaction is the generalization of
constraint satisfaction that allows for both universal and existential
quantifiers over constrained variables, instead
of just existential quantifiers.
We study quantified constraint satisfaction problems ${\rm CSP}(Q,S)$, where $Q$ denotes
a pattern of quantifier alternation ending in exists or the set of all possible
alternations of quantifiers, and $S$ is a set of relations constraining
the combinations of values that the variables may take.
These problems
belong to the corresponding level of the polynomial hierarchy or in PSPACE,
depending on whether $Q$ is a fixed pattern of quantifier alternation
or the set of all possible alternations of quantifiers.
We also introduce and study the
quantified constraint satisfaction problems ${\rm CSP'}(Q,S)$
in which the universally quantified variables are restricted to range
over given subsets of the domain.
We first show that ${\rm CSP}(Q,S)$ and ${\rm CSP'}(Q,S)$
are polynomial-time equivalent to
the problem of evaluating certain syntactically restricted
monadic second-order formulas on finite structures.
After this, we establish three broad sufficient conditions for polynomial-time
of ${\rm CSP'}(Q,S)$ that are based on closure functions; these results
generalize and extend earlier results by other researchers
about polynomial-time solvability of ${\rm CSP}(Q,S)$.
Our study culminates with a
dichotomy theorem for the complexity of list ${\rm CSP'}(Q,S)$,
that is, ${\rm CSP'}(Q,S)$ where the relations of $S$ include
every subset of the domain of $S$. Specifically, list
${\rm CSP'}(Q,S)$ is either solvable in polynomial-time or complete
for the corresponding level of the polynomial hierarchy, if
$Q$ is a fixed pattern of quantifier alternation (or PSPACE-complete
if $Q$ is the set of all possible alternations of quantifiers).

The proofs are based on a more general unique sink property formulation.

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