Marlene Gründel

Robert Ganian, Marlene Gründel, Simon Wietheger

Pearl's Causal Hierarchy (PCH) is a central framework for reasoning about probabilistic, interventional, and counterfactual statements, yet the satisfiability problem for PCH formulas is computationally intractable in almost all classical settings. We revisit this challenge through the lens of parameterized complexity and identify the first gateways to tractability. Our results include fixed-parameter and XP-algorithms for satisfiability in key probabilistic and counterfactual fragments, using parameters such as primal treewidth and the number of variables, together with matching hardness results that map the limits of tractability. Technically, we depart from the dynamic programming paradigm typically employed for treewidth-based algorithms and instead exploit structural characterizations of well-formed causal models, providing a new algorithmic toolkit for causal reasoning.

Robert Ganian, Marlene Gründel

Treewidth is a well-studied decompositional parameter to measure the tree-likeness of a graph. While the propositional satisfiability problem (SAT) is known to be tractable when parameterized by the treewidth of the underlying primal graph, the evaluation of quantified Boolean formulas (QBFs) remains PSPACE-complete even on formulas of constant treewidth. Intuitively, this is because ordinary treewidth does not take into account the prefix of the QBF: it neither distinguishes between existential and universal variables, nor accounts for the order in which they are quantified. In the past, several weaker variants of treewidth have been devised to incorporate prefix-sensitive information. To establish tractability for QBFs under these notions, prior work has employed either strategy- or resolution-based techniques, thereby dividing the parameterized complexity landscape of QBF into two regimes that are incomparable in strength. We establish fixed-parameter tractability with respect to bilateral treewidth, a novel and strictly more powerful decompositional parameter that combines these rivaling approaches by simultaneously allowing for branching on strategies and performing Q-resolution. As in previous works in this direction, our algorithm assumes that a suitable tree decomposition is provided on the input.