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

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 Ilayesian networks arguably the most central class of graphical models especially in what is known as the 50010 1)050(1 setting. A successful generic approach to optimal Bayesian network structure learning (BNSL), based on integer programming (IP); is implemented in the GOBNLP System. Despite the recent algoirithmic! 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 elliciemiT 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 N P-hard; (ii) We formalise mid analyse the relationship between three key polytopes underlynig 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.