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Complexity of QBF on Bounded Treedepth

Determine the computational complexity of Quantified Boolean Formulas (QBF) restricted to conjunctive normal form instances whose primal graphs have bounded treedepth; in particular, ascertain whether QBF on bounded-treedepth instances is polynomial-time solvable or remains Pspace-hard.

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Background

The paper leverages Atserias and Oliva’s result showing that QBF in CNF is Pspace-complete even on bounded pathwidth, and relates bounded pathwidth and treedepth to forbidden subgraph characterizations. While bounded pathwidth hardness is established, the analogous question for bounded treedepth remains unsettled and is recognized in the literature as a famous open problem.

Resolving this would clarify the boundary between tractability and hardness for quantified reasoning on structurally constrained instances and would inform similar questions for QCSP variants examined in the paper.

References

The complexity of QBF on bounded treedepth is a famous open problem (see e.g., where it is proved to be in P under some further restrictions).

— Graph Homomorphism, Monotone Classes and Bounded Pathwidth (2403.00497 - Eagling-Vose et al., 1 Mar 2024) in Introduction (Section 1)