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 complexity of the Markov equivalence criterion for ancestral graphs originally proposed by Zhao et al. (2005). While Zhao’s criterion is theoretically sound—it states that two ancestral graphs are Markov equivalent if and only if they share the same set of minimal collider paths and discriminating paths—the authors demonstrate that the number of such features can grow exponentially with the number of vertices in the graph. The authors begin by reviewing the definitions of ancestral graphs, Markov equivalence, and the specific path‑based features used in Zhao’s test. They then construct a family of graphs, essentially a bipartite core augmented with directed edges, for which the number of minimal collider paths and discriminating paths is on the order of 2^n, where n is the number of vertices. Using an inductive construction, they prove that each new vertex added to the graph introduces a combinatorial explosion of new, distinct paths, and that these paths are independent in the sense required by the equivalence test. This rigorous combinatorial argument shows that any algorithm that explicitly enumerates all required paths must, in the worst case, run in exponential time. The paper discusses the practical implications: existing algorithms based on Zhao’s criterion may become infeasible for moderately sized graphs because they would need to generate and compare an exponential number of path structures. The authors suggest that for restricted subclasses of ancestral graphs—such as trees, graphs with bounded degree, or those lacking certain edge configurations—a polynomial‑time equivalence test may still be possible. They conclude by outlining future research directions, including the design of alternative criteria that avoid exponential blow‑up, establishing tight lower bounds for the equivalence problem, and empirical studies to assess how often the worst‑case exponential behavior occurs in real‑world causal models. Overall, the work clarifies a fundamental limitation of the current Markov equivalence framework and points toward more scalable solutions for causal inference.