On the Edges of Characteristic Imset Polytopes
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The edges of the characteristic imset polytope, 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 CIM_G, the convex hull of the characteristic imsets of DAGs with skeleton G. We give a full edge-description of 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 CIM_G when G is a tree. Building on this connection we are also able to give a description of all edges of 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.
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