The Complexity of Quantified Constraint Satisfaction Problems under Structural Restrictions

We give a clear picture of the tractability/intractability frontier for quantified constraint satisfaction problems (QCSPs) under structural restrictions. On the negative side, we prove that checking QCSP satisfiability remains PSPACE-hard for all known structural properties more general than bounded treewidth and for the incomparable hypergraph acyclicity. Moreover, if the domain is not fixed, the problem is PSPACE-hard even for tree-shaped constraint scopes. On the positive side, we identify relevant tractable classes, including QCSPs with prefix ∃∀ having bounded hypertree width, and QCSPs with a bounded number of guards. The latter are solvable in polynomial time without any bound on domains or quantifier alternations.