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The challenging task of learning structures of probabilistic graphical models is an important problem within modern AI research. Recent years have witnessed several major algorithmic advances in structure  learning for Bayesian networks - arguably the most central class of graphical models - especially in what is known as the  score-based setting. A successful generic approach to optimal Bayesian network structure learning (BNSL), based on  integer programming (IP), is implemented in the GOBNILP system.  Despite the recent algorithmic advances, current understanding of foundational aspects underlying the IP based approach to BNSL is still somewhat lacking.  Understanding fundamental aspects of cutting planes and the related separation problem is important not only from a purely theoretical perspective, but also since it holds out the promise of further improving the efficiency of state-of-the-art approaches to solving BNSL exactly.  In this paper, we make several theoretical contributions towards these goals: (i) we study the computational complexity of the separation problem, proving that the problem is NP-hard; (ii) we formalise and analyse the relationship between three key polytopes underlying the IP-based approach to BNSL; (iii) we study the facets of the three polytopes both from the theoretical and practical perspective, providing, via exhaustive computation, a complete enumeration of facets for low-dimensional family-variable polytopes; and, furthermore, (iv) we establish a tight connection of the BNSL problem to the acyclic subgraph problem.

Bayesian Network Structure Learning with Integer Programming: Polytopes, Facets and Complexity