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.

翻译：因果有向无环图（也称为贝叶斯网络）是编码随机变量之间条件依赖关系的流行工具。在因果DAG中，随机变量被建模为DAG中的顶点，并规定每个随机变量在其父节点条件下独立于其祖先节点。然而，对于同一组随机变量，两个不同的因果DAG可能编码完全相同的条件依赖关系集。这样的因果DAG被称为马尔可夫等价的，而马尔可夫等价DAG的等价类称为马尔可夫等价类。过去几十年中，人们对MEC的优美组合刻画进行了深入研究，特别地，已知同一MEC中的所有DAG必须具有相同的"骨架"（底层无向图）和v-结构（形如$a\rightarrow b \leftarrow c$的诱导子图）。这些组合刻画也提出了若干自然的算法问题。其中之一是：给定一个无向图$G$作为输入，有多少个不同的马尔可夫等价类具有骨架$G$？近年来，大量工作致力于解决这一问题及其他密切相关的课题。然而，据我们所知，该问题的多项式时间算法仍未可知。在本文中，我们通过给出上述问题的一个固定参数可解算法向此目标迈进，其中参数为输入图$G$的树宽和最大度。我们工作的主要技术成分是一个称为"影子"的构造，它使我们能够对MEC组合刻画所施加的长程约束进行"局部描述"。