Parameterized Algorithms for Bayesian Network Learning

Abstract

We investigate the parameterized complexity of Bayesian Network Structure Learning (BNSL), a classical problem that has received significant attention in empirical but also purely theoretical studies. We follow up on previous works that have analyzed the complexity of BNSL w.r.t. the so-called superstructure of the input. While known results imply that BNSL is unlikely to be fixed-parameter tractable even when parameterized by the size of a vertex cover in the superstructure, here we show that a different kind of parameterization—notably by the size of a feedback edge set—yields fixed-parameter tractability. We proceed by showing that this result can be strengthened to a localized version of the feedback edge set. We adapt corresponding algorithms to the closely related problem of Polytree Learning. Concerning the lower bounds, we establish W[1]-hardness of BNSL parameterized by tree-cut width.We then analyze how the complexity of BNSL depends on the representation of the input. In particular, while the bulk of past theoretical work on the topic assumed the use of the so-called non-zero representation, here we prove that if an additive representation can be used instead then BNSL becomes fixed-parameter tractable even under significantly milder restrictions to the superstructure, notably when parameterized by the treewidth alone.