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# The Chordal Graph Polytope for Learning Decomposable Models

*Proceedings of the Eighth International Conference on Probabilistic Graphical Models*, PMLR 52:499-510, 2016.

#### Abstract

This theoretical paper is inspired by an \em integer linear programming (ILP) approach to learning the structure of \em decomposable models. We intend to represent decomposable models by special zero-one vectors, named \em characteristic imsets. Our approach leads to the study of a special polytope, defined as the convex hull of all characteristic imsets for chordal graphs, named the \em chordal graph polytope. We introduce a class of \em clutter inequalities and show that all of them are valid for (the vectors in) the polytope. In fact, these inequalities are even facet-defining for the polytope and we dare to conjecture that they lead to a complete polyhedral description of the polytope. Finally, we propose an LP method to solve the \em separation problem with these inequalities for use in a cutting plane approach.