Polynomial-time identification in linear SEMs with known graphs

Determine whether a polynomial-time algorithm exists for identifying interventional queries in linear structural equation models when the causal graph is known, analogous to the polynomial-time ID algorithm available for nonparametric models.

Background

Symbolic algorithms such as ID provide sound and complete, polynomial-time identification for interventional queries in nonparametric graphical models. For linear structural equation models, identification often reduces to solving systems of polynomial equations, but the computational complexity of a general identification algorithm remains unresolved.

The authors explicitly state that it is currently unknown whether an efficient (polynomial-time) algorithm exists for the linear case when the causal graph is known.

References

For example, when the causal graph is known, the identification of interventional queries can be solved in polynomial time in nonparametric models; however, at the time of writing, whether such an algorithm exists for linear models remains an open question.

— Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference (2508.17099 - Cinelli et al., 23 Aug 2025) in Section: Automation of the Causal Inference Pipeline — Hardness results