A Refined View of Causal Graphs and Component Sizes: SP-Closed Graph Classes and Beyond

The causal graph of a planning instance is an important tool for planning both in practice and in theory. The theoretical studies of causal graphs have largely analysed the computational complexity of planning for instances where the causal graph has a certain structure, often in combination with other parameters like the domain size of the variables. Chen and Gimand#233;nez ignored even the structure and considered only the size of the weakly connected components. They proved that planning is tractable if the components are bounded by a constant and otherwise intractable. Their intractability result was, however, conditioned by an assumption from parameterised complexity theory that has no known useful relationship with the standard complexity classes. We approach the same problem from the perspective of standard complexity classes, and prove that planning is NP-hard for classes with unbounded components under an additional restriction we refer to as SP-closed. We then argue that most NP-hardness theorems for causal graphs are difficult to apply and, thus, prove a more general result; even if the component sizes grow slowly and the class is not densely populated with graphs, planning still cannot be tractable unless the polynomial hierachy collapses. Both these results still hold when restricted to the class of acyclic causal graphs. We finally give a partial characterization of the borderline between NP-hard and NP-intermediate classes, giving further insight into the problem.

💡 Research Summary

The paper revisits the long‑standing question of how the structure of a planning problem’s causal graph influences its computational difficulty, but it does so from the perspective of classical complexity theory rather than parameterised complexity. A causal graph captures the dependencies among state variables and actions; its shape has been used extensively to derive tractability results (e.g., bounded tree‑width, bounded indegree, limited domain size). Chen and Giménez previously showed that if the weakly connected components of the causal graph are bounded by a constant then planning is polynomial‑time solvable, while unbounded component size leads to “intractability”. Their hardness proof, however, relies on a W