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Bayesian network structure learning with integer programming: polytopes, facets and complexity.

Cussens, James; J�rvisalo, Matti; Korhonen, Janne H.; Bartlett, Mark


James Cussens

Matti J�rvisalo

Janne H. Korhonen


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.


CUSSENS, J., JÄRVISALO, M., KORHONEN, J.H. and BARTLETT, M. 2017. Bayesian network structure learning with integer programming: polytopes, facets and complexity. Journal of artificial intelligence research [online], 58, pages 185-229. Available from:

Journal Article Type Article
Acceptance Date May 31, 2016
Online Publication Date Jan 26, 2017
Publication Date Jan 31, 2017
Deposit Date Jan 20, 2020
Publicly Available Date Jan 20, 2020
Journal Journal of Artificial Intelligence Research
Print ISSN 1076-9757
Publisher AI Access Foundation
Peer Reviewed Peer Reviewed
Volume 58
Pages 185-229
Keywords Learning structures; Artificial intelligence; Bayesian networks; Cutting planes; Separation problem; Probabilistic graphic models
Public URL


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