We study the problem of learning the structure of an optimal Bayesian network when additional constraints are posed on the network or on its moralized graph. More precisely, we consider the constraint that the network or its moralized graph are close, in terms of vertex or edge deletions, to a sparse graph class $\Pi$. For example, we show that learning an optimal network whose moralized graph has vertex deletion distance at most $k$ from a graph with maximum degree 1 can be computed in polynomial time when $k$ is constant. This extends previous work that gave an algorithm with such a running time for the vertex deletion distance to edgeless graphs [Korhonen & Parviainen, NIPS 2015]. We then show that further extensions or improvements are presumably impossible. For example, we show that learning optimal networks where the network or its moralized graph have maximum degree $2$ or connected components of size at most $c$, $c\ge 3$, is NP-hard. Finally, we show that learning an optimal network with at most $k$ edges in the moralized graph presumably has no $f(k)\cdot |I|^{O(1)}$-time algorithm and that, in contrast, an optimal network with at most $k$ arcs can be computed in $2^{O(k)}\cdot |I|^{O(1)}$ time where $|I|$ is the total input size.

翻译：我们研究在网络或道德化图中出现额外限制时学习最佳巴伊西亚网络结构的问题。 更确切地说, 我们考虑的是网络或其道德化图在顶点或边缘删除方面接近一个稀薄的图形类$\Pi$。 例如, 我们显示, 学习一个其道德化图在最高水平1的图表中以最高水平1美元删除距离的最理想网络, 在美元不变的情况下, 可以以多元值美元计算其最高水平1美元。 这扩展了以前的工作, 使一个算法在顶点删除距离到无边缘的图表[ Korhoonen & Parviainen, NIPS 2015] 的运行时间。 然后我们显示, 进一步的扩展或改进大概是不可能的。 例如, 我们显示, 在网络或其道德化图中, 最高水平为$2美元或最多规模的相连接的网络, 3美元, 硬度。 最后, 我们显示, 在道德化图中, 在最高水平化图中, $%I\\\\\\\\ xO 里, 在最高水平的轨道上, 可以进行最优的网络比较。