Abstract:

Efficiently counting Markov equivalent directed acyclic graphs (DAGs) is crucial in graphical causal analysis. Wienöbst et al. (2023) introduced a polynomial-time algorithm, known as the Clique-Picking algorithm, to count the number of Markov equivalent DAGs for a given completed partially directed acyclic graph (CPDAG). This algorithm iteratively selects a root clique, determines fixed orientations with outgoing edges from the clique, and generates the unresolved undirected connected components (UCCGs). In this work, we propose a more efficient approach to UCCG generation by utilizing previously computed results for different root cliques. Our method introduces the concept of super cliques within rooted clique trees, enabling their efficient transfer between trees with different root cliques. The proposed algorithm effectively reduces the computational complexity of the Clique-Picking method, particularly when the number of cliques is substantially smaller than the number of vertices and edges.

Lay Summary: In the study of causal inference, Directed Acyclic Graphs (DAGs) are fundamental tools for representing cause-and-effect relationships among variables. However, observational data alone often cannot identify a unique DAG. Instead, we can only determine a **Markov equivalence class (MEC)**—a set of DAGs that encode the same conditional independence relations. Understanding the size of a MEC, i.e., the number of DAGs it contains, is crucial for evaluating causal uncertainty and designing informative interventions. Yet, counting the number of DAGs in a MEC is computationally challenging, particularly as the size and complexity of the graph grow.To address this, the **Clique-Picking algorithm** (Wienöbst et al., 2023) was recently proposed as an efficient polynomial-time method for MEC size computation. This method explores different cliques as roots in a clique tree and recursively computes the connected components of oriented subgraphs. While effective, this approach still incurs high computational cost due to repeated processing of similar graph structures when different root cliques are selected.This paper presents an improved solution by introducing the concept of **super cliques** and developing a new **Super Cliques Transfer Algorithm**. Super cliques capture groups of related cliques in the clique tree and allow the reuse of previously computed results. The algorithm exploits overlapping structures among different root selections, enabling efficient transfer of information between clique trees rooted at different cliques.As a result, the proposed method significantly reduces the computational complexity of a key step in the original Clique-Picking algorithm—from $\mathcal{O}(m(|V|+|E|))$ to $\mathcal{O}(m^2)$, where $m$ is the number of cliques. The paper further provides both intuitive (tree-based) and theoretical (sequence-based) formulations of super cliques, which support the algorithm's correctness and generalizability.Extensive experiments demonstrate substantial speedup of the improved method (referred to as ICP) over the original algorithm, especially for large or dense graphs. This advancement makes MEC size computation more practical for high-dimensional causal analysis tasks and opens up broader applications in fields such as epidemiology, policy modeling, and automated reasoning.