Towards Counting Markov Equivalence Classes with Logical Constraints

TL;DR

This paper develops a polynomial-time algorithm for enumerating unique causal graph structures under logical constraints, advancing understanding of the complexity of causal inference in DAG models.

Contribution

It introduces a novel approach combining first-order logic with combinatorics to analyze MECs of size one efficiently.

Findings

Polynomial-time enumeration of essential DAGs with logical constraints

Integration of first-order model counting with MEC analysis

Provides tools for understanding causal inference complexity

Abstract

We initiate the study of counting Markov Equivalence Classes (MEC) under logical constraints. MECs are equivalence classes of Directed Acyclic Graphs (DAGs) that encode the same conditional independence structure among the random variables of a DAG model. Observational data can only allow to infer a DAG model up to Markov Equivalence. However, Markov equivalent DAGs can represent different causal structures, potentially super-exponentially many. Hence, understanding MECs combinatorially is critical to understanding the complexity of causal inference. In this paper, we focus on analysing MECs of size one, with logical constraints on the graph topology. We provide a polynomial-time algorithm (w.r.t. the number of nodes) for enumerating essential DAGs (the only members of an MEC of size one) with arbitrary logical constraints expressed in first-order logic with two variables and counting quantifiers (C^2). Our work brings together recent developments in tractable first-order model counting and combinatorics of MECs.