Efficient Enumeration of Markov Equivalent DAGs

TL;DR

This paper introduces the first linear-time delay algorithm for enumerating all DAGs in a Markov equivalence class, significantly improving computational efficiency in causal analysis.

Contribution

It presents a novel linear-time delay algorithm for enumerating Markov equivalent DAGs and extends it to models with background knowledge like MPDAGs.

Findings

The algorithm achieves linear delay in enumeration.

All MEC members can be listed with successive DAGs differing by at most three edges.

The implementation is efficient and validated through experiments.

Abstract

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.