Efficient Enumeration of Markov Equivalent DAGs
This addresses a computational bottleneck in causal analysis for researchers and practitioners, representing a strong specific gain rather than a foundational breakthrough.
The paper tackles the problem of enumerating directed acyclic graphs (DAGs) in a Markov equivalence class with high computational efficiency, achieving the first linear-time delay algorithm and showing that successive DAGs can have a structural Hamming distance of at most three.
Enumerating the directed acyclic graphs (DAGs) of a Markov equivalence class (MEC) is an important primitive in causal analysis. The central resource from the perspective of computational complexity is the delay, that is, the time an algorithm that lists all members of the class requires between two consecutive outputs. Commonly used algorithms for this task utilize the rules proposed by Meek (1995) or the transformational characterization by Chickering (1995), both resulting in superlinear delay. In this paper, we present the first linear-time delay algorithm. On the theoretical side, we show that our algorithm can be generalized to enumerate DAGs represented by models that incorporate background knowledge, such as MPDAGs; on the practical side, we provide an efficient implementation and evaluate it in a series of experiments. Complementary to the linear-time delay algorithm, we also provide intriguing insights into Markov equivalence itself: All members of an MEC can be enumerated such that two successive DAGs have structural Hamming distance at most three.