Polynomial-Time Algorithms for Counting and Sampling Markov Equivalent DAGs

Abstract

Counting and uniform sampling of directed acyclic graphs (DAGs) from a Markov equivalence class are fundamental tasks in graphical causal analysis. In this paper, we show that these tasks can be performed in polynomial time, solving a long-standing open problem in this area. Our algorithms are effective and easily implementable. Experimental results show that the algorithms significantly outperform state-of-the-art methods.

Practical Algorithms for Orientations of Partially Directed Graphical Models

“…While the algorithms for computing maximal orientations used in practice resort to directly applying the Meek rules, the best theoretical methods are based on two other important primitives of causal analysis: (i) the consistent extension of a PDAG to a DAG and (ii) computing the CPDAG representation of the graphs Markov equivalent to a given DAG. This was already mentioned by Chickering (1995) and generalized to instances with background knowledge by Wienöbst et al (2021). Using the clever algorithm of Chickering (1995), the second task (ii) can be solved in linear time.…”

“…As discussed above, algorithms for extending PDAGs play a key role in the maximal orientation task. While, from a theoretical point of view, previous results suggest that it is likely not possible to further improve the asymptotic run time of extendability algorithms, as Wienöbst et al (2021) gave a conditional O(n 3 ) lower bound for combinatorial algorithms, from a practical perspective, the current algorithms are either extremely simple, as Dor-Tarsi's approach, or very sophisticated with considerable practical overhead, as the methods proposed by Wienöbst et al (2021). In this work, we are searching for a middle ground and give two novel approaches to extend PDAGs, with the main focus lying on simplicity and effectiveness.…”

“…The primary significance of the MPDAG model, including completed PDAGs (CPDAGs, also called essential graphs) (Andersson et al, 1997) is that it represents Markov equivalent DAGs in an elegant and convenient way. It is commonly used for solving many important problems as, e.g., counting and sampling Markov equivalent DAGs (He et al, 2015;Wienöbst et al, 2021), estimating causal effects (Maathuis et al, 2009;van der Zander and Liśkiewicz, 2016;Perković et al, 2017), or learning causal models (Chickering, 2002), where CPDAGs represent the states of the search space. Consequently, orienting a given PDAG maximally, is a frequently used primitive in causal inference and discovery, perhaps most prominently arising in constraint-based causal structure learning, as, e. g., the final step in the PC algorithm (Spirtes et al, 2000;Kalisch and Bühlman, 2007) and its modifications.…”

“…This approach is used in the GES algorithm (Chickering, 2002) and in a modified way in the GIES algorithm (Hauser and Bühlmann, 2012), the latter however having the same worst-case complexity as the direct application of the Meek rules discussed above. More recently, Wienöbst et al (2021) generalized both approaches, while maintaining the optimal time complexity of O(n 3 ), to maximally orient a PDAG to an MPDAG.…”

“…On the other hand, we aim to illustrate the strengths of using consistent DAG extensions in the computation of the maximal orientation of a PDAG. While this approach has been proposed theoretically in (Chickering, 1995;Wienöbst et al, 2021), it has only been used in special cases and has not found widespread application in practice, evidenced by the fact that most software packages such as pcalg (Kalisch et al, 2012) and causaldag (Squires, 2018) rely on algorithms, which apply the Meek rules directly. Utilizing consistent DAG extensions instead, yields a better worst-case complexity and is superior in practice as we demonstrate in our experiments.…”