On the Gröbner fans of max-linear Bayesian networks

Max-linear Bayesian networks (MLBN) are statistical models described by weighted directed acyclic graphs (weighted DAG). At the same time, we can associate to this weighted DAG an alcoved polyhedron, a polyhedron whose facet normals correspond to the type $A_n$ root system. This alcoved polyhedron also describes the feasible set of an optimal transport problem on the same weighted DAG. The combinatorics of this optimal transport problem are encoded by the Gröbner fan. We use this fact to enumerate the combinatorial types of alcoved polyhedra associated to MLBN