A polynomial-time exact algorithm for counting the number of directed acyclic graphs in a Markov equivalence class was recently given by Wien\"obst, Bannach, and Li\'skiewicz (AAAI 2021). In this paper, we consider the more general problem of counting the number of directed acyclic graphs in a Markov equivalence class when the directions of some of the edges are also fixed (this setting arises, for example, when interventional data is partially available). This problem has been shown in earlier work to be complexity-theoretically hard. In contrast, we show that the problem is nevertheless tractable in an interesting class of instances, by establishing that it is ``fixed-parameter tractable''. In particular, our counting algorithm runs in time that is bounded by a polynomial in the size of the graph, where the degree of the polynomial does \emph{not} depend upon the number of additional edges provided as input.

翻译：最近，Wienöbst、Bannach 和 Liśkiewicz（AAAI 2021）提出了一种多项式时间精确算法，用于计算马尔可夫等价类中有向无环图的数量。本文考虑更一般的问题：当部分边的方向也被固定时（例如，在部分干预数据可用的情况下），如何计算马尔可夫等价类中有向无环图的数量。早期研究表明，该问题在复杂性理论上具有困难性。相反，我们证明该问题在一类有趣的实例中仍然是可处理的，即它具有“固定参数可处理性”。特别地，我们的计数算法运行时间在图的规模上呈多项式级数，且多项式的度数不依赖于作为输入提供的附加边的数量。