The edges of the characteristic imset polytope, $\operatorname{CIM}_p$, were recently shown to have strong connections to causal discovery as many algorithms could be interpreted as greedy restricted edge-walks, even though only a strict subset of the edges are known. To better understand the general edge structure of the polytope we describe the edge structure of faces with a clear combinatorial interpretation: for any undirected graph $G$ we have the face $\operatorname{CIM}_G$, the convex hull of the characteristic imsets of DAGs with skeleton $G$. We give a full edge-description of $\operatorname{CIM}_G$ when $G$ is a tree, leading to interesting connections to other polytopes. In particular the well-studied stable set polytope can be recovered as a face of $\operatorname{CIM}_G$ when $G$ is a tree. Building on this connection we are also able to give a description of all edges of $\operatorname{CIM}_G$ when $G$ is a cycle, suggesting possible inroads for generalization. We then introduce an algorithm for learning directed trees from data, utilizing our newly discovered edges, that outperforms classical methods on simulated Gaussian data.

翻译：显性 聚点的边缘, $\ operatorname{ CIM ⁇ p$, 最近被证明与因果关系发现有着密切的联系, 因为许多算法可以被解释为贪婪的边行限制边行, 即使只知道边缘的严格子集。 为了更好地了解多点的普通边缘结构, 我们用清晰的组合解释来描述面的边缘结构: 对于任何未引导的图形 $G$, 我们拥有面值$\ operatorname{ CIM ⁇ G$, 与骨架 DAG 特征的顶层相近。 当$G$是一棵树时, 我们给出了$\ operatorname{ CIM ⁇ G$的完全边界描述, 当$G$是一棵树树时, 我们的稳定的多点结构可以恢复为$@CIM ⁇ G$的面值。 在此连接上, 我们也可以描述 $$Operatorname{C} G$ 完全的边框。 当我们从 Sqolentalal 中学习了一种新数据时, 我们的G 正在学习一个 Grodeal_ 。