The structure of Markov equivalence classes (MECs) of causal DAGs has been studied extensively. A natural question in this regard is to algorithmically find the number of MECs with a given skeleton. Until recently, the known results for this problem were in the setting of very special graphs (such as paths, cycles, and star graphs). More recently, a fixed-parameter tractable (FPT) algorithm was given for this problem which, given an input graph $G$, counts the number of MECs with the skeleton $G$ in $O(n(2^{O(d^4k^4)} + n^2))$ time, where $n$, $d$, and $k$, respectively, are the numbers of nodes, the degree, and the treewidth of $G$. We give a faster FPT algorithm that solves the problem in $O(n(2^{O(d^2k^2)} + n^2))$ time when the input graph is chordal. Additionally, we show that the runtime can be further improved to polynomial time when the input graph $G$ is a tree.

翻译：因果有向无环图的马尔可夫等价类（MECs）结构已被广泛研究。在此背景下，一个自然问题是算法性地找出具有给定骨架的MECs数量。直到最近，该问题的已知结果仅适用于非常特殊的图（如路径、环和星形图）。近期，针对该问题提出了一种固定参数可处理（FPT）算法，该算法对于输入图$G$，能在$O(n(2^{O(d^4k^4)} + n^2))$时间内计数骨架为$G$的MECs数量，其中$n$、$d$、$k$分别表示$G$的节点数、度数和树宽。我们提出了一种更快的FPT算法，当输入图为弦图时，能在$O(n(2^{O(d^2k^2)} + n^2))$时间内解决该问题。此外，我们证明当输入图$G$为树时，运行时间可进一步改进为多项式时间。