Supplementary material for Markov equivalence for ancestral graphs

We prove that the criterion for Markov equivalence provided by Zhao et al. (2005) may involve a set of features of a graph that is exponential in the number of vertices.

💡 Research Summary

The paper investigates the computational feasibility of the Markov equivalence criterion for maximal ancestral graphs (MAGs) introduced by Zhao, Zheng, and Liu in 2005. Their theorem states that two MAGs are Markov equivalent if and only if they share exactly the same set of minimal collider paths. A minimal collider path is a collider‑structured path that cannot be decomposed into shorter collider paths, and the collection of such paths uniquely characterizes the conditional independence structure of a MAG. While elegant, the practical applicability of this characterization hinges on the size of the minimal collider path set.

To expose the potential combinatorial explosion, the authors construct a family of bi‑directed bipartite graphs (G_k). The vertex set is partitioned into (V_1) (k vertices) and (V_2) (one vertex (v_{ab}) for each unordered pair ({a,b}) with (a\neq b) in (V_1)). Edges are only of the form (a \leftrightarrow v_{ab}) and (b \leftrightarrow v_{ab}); there are no edges within (V_1) or within (V_2). Consequently each vertex in (V_1) has degree (k-1) and each vertex in (V_2) has degree 2.

Given any ordered permutation ((a_1, a_2, \dots, a_k)) of the vertices in (V_1), one can form a path
(a_1 \leftrightarrow v_{a_1a_2} \leftrightarrow a_2 \leftrightarrow v_{a_2a_3} \leftrightarrow \dots \leftrightarrow a_{k-1} \leftrightarrow v_{a_{k-1}a_k} \leftrightarrow a_k).
All interior vertices are colliders, making the whole sequence a collider path. Because no proper subsequence of this path is itself a collider path, it is minimal. Since there are (k!) permutations of (V_1), the graph (G_k) contains at least (k!/2) distinct minimal collider paths (the reverse of a permutation yields the same undirected path).

The total number of vertices in (G_k) is (|V_1| + |V_2| = k + \binom{k}{2} = k(k+1)/2). For any fixed (n), define (k(n)) as the largest integer satisfying (k(k+1)/2 \le n). One can show (k(n) \ge \sqrt{n}). Hence a graph with at most (n) vertices can still possess on the order of ((\sqrt{n})! /2) minimal collider paths, which grows faster than any polynomial in (n).

This construction demonstrates that the Zhao‑Zheng‑Liu criterion may require examining a set of features whose size is exponential in the number of vertices. Consequently, directly enumerating or comparing minimal collider paths is computationally infeasible for large graphs, despite the theoretical completeness of the criterion. The paper thus calls for alternative algorithms that either compress the set of minimal collider paths or replace the criterion with a polynomial‑time test, highlighting an important gap between theoretical characterizations of Markov equivalence and their practical implementation in causal discovery and graphical modeling.