Complexity of QBF on bounded treedepth

Determine the computational complexity of Quantified Boolean Formulas (QBF) when restricted to instances whose primal graphs have bounded treedepth; in particular, ascertain whether QBF is polynomial-time decidable on this structural restriction.

Background

The authors use a result of Atserias and Oliva to show Pspace-completeness of bounded-pathwidth QBF and leverage this to prove related hardness results. They highlight that moving from bounded pathwidth to bounded treedepth dramatically changes the landscape and that the complexity of QBF under bounded treedepth is a well-known unresolved question.

This open problem is central to understanding the limits of structural decompositions that enforce extreme sparsity and depth constraints and has implications for related quantified problems, including QCSP variants discussed in the paper.