A Fixed-Parameter Tractable Algorithm for Counting Markov Equivalence Classes with the same Skeleton

TL;DR

This work addresses counting Markov equivalence classes (MECs) with a given skeleton in causal graphs. It introduces a fixed-parameter tractable approach parameterized by treewidth $k$ and maximum degree $\delta$, achieving a runtime of $O(n(2^{O(k^4\delta^4)}+n^2))$ by partitioning MECs via shadows and using projections to combine subproblem solutions along tree decompositions. The core innovations are the shadow concept, the derived path function (DPF), and a tailored LBFS construction that respects the shadow constraints to realize consistent MECs. The methodology yields a principled, modular dynamic-programming framework for MEC counting that scales polynomially in the input size for fixed $k$ and $\delta$, representing a significant step toward understanding the computational complexity of MEC counting. The results also situate this problem within a broader fixed-parameter paradigm for graphical models, suggesting paths to potential polynomial-time algorithms or hardness proofs in future work.

Abstract

Causal DAGs (also known as Bayesian networks) are a popular tool for encoding conditional dependencies between random variables. In a causal DAG, the random variables are modeled as vertices in the DAG, and it is stipulated that every random variable is independent of its ancestors conditioned on its parents. It is possible, however, for two different causal DAGs on the same set of random variables to encode exactly the same set of conditional dependencies. Such causal DAGs are said to be Markov equivalent, and equivalence classes of Markov equivalent DAGs are known as Markov Equivalent Classes (MECs). Beautiful combinatorial characterizations of MECs have been developed in the past few decades, and it is known, in particular that all DAGs in the same MEC must have the same "skeleton" (underlying undirected graph) and v-structures (induced subgraph of the form $a\rightarrow b \leftarrow c$). These combinatorial characterizations also suggest several natural algorithmic questions. One of these is: given an undirected graph $G$ as input, how many distinct Markov equivalence classes have the skeleton $G$? Much work has been devoted in the last few years to this and other closely related problems. However, to the best of our knowledge, a polynomial time algorithm for the problem remains unknown. In this paper, we make progress towards this goal by giving a fixed parameter tractable algorithm for the above problem, with the parameters being the treewidth and the maximum degree of the input graph $G$. The main technical ingredient in our work is a construction we refer to as shadow, which lets us create a "local description" of long-range constraints imposed by the combinatorial characterizations of MECs.

Figures (3)

• Figure 1: Markov equivalent DAGs and MECs with the same skeleton. $D_1, D_2$, and $D_3$ are Markov equivalent, while $D_4$ is not equivalent to $D_1, D_2$, and $D_3$. The MEC $M_1 = \{D_1, D_2, D_3\}$ contains $D_1, D_2$ and $D_3$, and its graphical representation is the union of $D_1, D_2$, and $D_3$. The MEC $M_2$ contains only $D_4$, and its graphical representation matches $D_4$. The MECs $M_1$ and $M_2$ share $G$ as their skeleton, and in fact are the only MECs with skeleton $G$. Both $M_1$ and $M_2$ entail a conditional independence relation of the form $B \perp C \mid S$, where in $M_1$, $S = \{C\}$, and in $M_2$, $S = \varnothing$.
• Figure 2: Strongly protected $u \rightarrow v$.
• Figure 3: Example: $G$ is an undirected graph, and $T$ is a tree decomposition of $G$ where: $X_1 = \{1, 2, 3\}$, $X_2 = \{2, 4, 5\}$, $X_3 = \{2, 3, 6\}$, $X_4 = \{3, 7, 8\}$, $X_5 = \{5, 9, 10\}$, $X_6 = \{7, 11, 12\}$, $X_7 = \{8, 13, 14\}$, $X_8 = \{12, 15, 16\}$, and $X_9 = \{14, 17\}$. $T_4$, $T_6$, and $T_7$ are induced subtrees of $T$ rooted at $X_4$, $X_6$, and $X_7$, respectively. We assume $X_6$ and $X_7$ are the first and second children of $X_4$. $T_4^0$ is an induced subgraph of $T$ containing only the node $X_4$. $T_4^1$ is an induced subtree of $T$ containing node $X_4$ and the nodes of $T_6$, and $T_4^2$ is an induced subtree of $T_4$ containing node $X_4$ and the nodes of $T_6$ and $T_7$. $G_4$ is the induced subgraph of $G$ represented by $T_4$. $G_4^0$ is the induced subgraph of $G$ represented by $T_4^0$. $G_4^1$ is the induced subgraph of $G$ represented by $T_4^1$. $G_4^2$ is the induced subgraph of $G$ represented by $T_4^2$. Since $T_4 = T_4^2$, it follows that $G_4 = G_4^2$.

Theorems & Definitions (204)

• Definition 2.1: Path, chord of a path, descendant, parent
• Definition 2.2: Cycle, chord of a cycle, DAG
• Definition 2.3: Chordal Graph
• Definition 2.4: Chain Graph
• Proposition 2.5
• proof
• Proposition 2.6
• proof
• Proposition 2.7
• proof