Learning Sparse Causal Models is not NP-hard

Tom Claassen, Joris Mooij, Tom Heskes
Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence, PMLR R11:75-84, 2013.

Abstract

This paper shows that causal model discovery is not an NP-hard problem, in the sense that for sparse graphs bounded by node degree k the sound and complete causal model can be obtained in worst case order N 2(k+2) independence tests, even when latent variables and selection bias may be present. We present a modification of the well-known FCI algorithm that implements the method for an independence oracle, and suggest improvements for sample/real-world data versions. It does not contradict any known hardness results, and does not solve an NP-hard problem: it just proves that sparse causal discovery is perhaps more complicated, but not as hard as learning minimal Bayesian networks.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR11-claassen13a, title = {Learning Sparse Causal Models is not {NP}-hard}, author = {Claassen, Tom and Mooij, Joris and Heskes, Tom}, booktitle = {Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence}, pages = {75--84}, year = {2013}, editor = {Nicholson, Ann and Smyth, Padhraic}, volume = {R11}, series = {Proceedings of Machine Learning Research}, month = {12--14 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r11/main/assets/claassen13a/claassen13a.pdf}, url = {https://proceedings.mlr.press/r11/claassen13a.html}, abstract = {This paper shows that causal model discovery is not an NP-hard problem, in the sense that for sparse graphs bounded by node degree k the sound and complete causal model can be obtained in worst case order N 2(k+2) independence tests, even when latent variables and selection bias may be present. We present a modification of the well-known FCI algorithm that implements the method for an independence oracle, and suggest improvements for sample/real-world data versions. It does not contradict any known hardness results, and does not solve an NP-hard problem: it just proves that sparse causal discovery is perhaps more complicated, but not as hard as learning minimal Bayesian networks.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Learning Sparse Causal Models is not NP-hard %A Tom Claassen %A Joris Mooij %A Tom Heskes %B Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2013 %E Ann Nicholson %E Padhraic Smyth %F pmlr-vR11-claassen13a %I PMLR %P 75--84 %U https://proceedings.mlr.press/r11/claassen13a.html %V R11 %X This paper shows that causal model discovery is not an NP-hard problem, in the sense that for sparse graphs bounded by node degree k the sound and complete causal model can be obtained in worst case order N 2(k+2) independence tests, even when latent variables and selection bias may be present. We present a modification of the well-known FCI algorithm that implements the method for an independence oracle, and suggest improvements for sample/real-world data versions. It does not contradict any known hardness results, and does not solve an NP-hard problem: it just proves that sparse causal discovery is perhaps more complicated, but not as hard as learning minimal Bayesian networks. %Z Reissued by PMLR on 04 October 2026.
APA
Claassen, T., Mooij, J. & Heskes, T.. (2013). Learning Sparse Causal Models is not NP-hard. Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R11:75-84 Available from https://proceedings.mlr.press/r11/claassen13a.html. Reissued by PMLR on 04 October 2026.

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