Extendability of causal graphical models: Algorithms and computational complexity
Marcel Wienöbst, Max Bannach, Maciej Liśkiewicz
Abstract
Finding a consistent DAG extension for a given partially directed acyclic graph (PDAG) is a basic building block used in graphical causal analysis. In 1992, Dor and Tarsi proposed an algorithm with time complexity O(n^4), which has been widely used in causal theory and practice so far. It is a long-standing open question whether an extension can be computed faster and, in particular, it was conjectured that a linear-time method may exist. The main contributions of our work are two-fold: Firstly, we propose a new algorithm for the extension problem for PDAGs which runs in time O(n^3); secondly, we show that, under a computational intractability assumption, our cubic algorithm is optimal. Thus, our impossibility result disproves the conjecture that a linear-time method exists. Based on these results, we present a full complexity landscape for finding extensions in various causal graphical models. We extend the techniques to recognition problems and apply them to design an effective algorithm for closing a PDAG under the orientation rules of Meek.
BibTeX
@InProceedings{pmlr-v161-wienobst21a,
title = {Extendability of causal graphical models: Algorithms and computational complexity},
author = {Wien\"{o}bst, Marcel and Bannach, Max and Li\'{s}kiewicz, Maciej},
booktitle = {Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence},
pages = {1248--1257},
year = {2021},
editor = {de Campos, Cassio and Maathuis, Marloes H.},
volume = {161},
series = {Proceedings of Machine Learning Research},
month = {27--30 Jul},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v161/wienobst21a/wienobst21a.pdf},
url = {https://proceedings.mlr.press/v161/wienobst21a.html},
abstract = {Finding a consistent DAG extension for a given partially directed acyclic graph (PDAG) is a basic building block used in graphical causal analysis. In 1992, Dor and Tarsi proposed an algorithm with time complexity O(n^4), which has been widely used in causal theory and practice so far. It is a long-standing open question whether an extension can be computed faster and, in particular, it was conjectured that a linear-time method may exist. The main contributions of our work are two-fold: Firstly, we propose a new algorithm for the extension problem for PDAGs which runs in time O(n^3); secondly, we show that, under a computational intractability assumption, our cubic algorithm is optimal. Thus, our impossibility result disproves the conjecture that a linear-time method exists. Based on these results, we present a full complexity landscape for finding extensions in various causal graphical models. We extend the techniques to recognition problems and apply them to design an effective algorithm for closing a PDAG under the orientation rules of Meek.}
}