On Tractability of Learning Bayesian Networks with Ancestral Constraints

Expert knowledge can greatly reduce the complexity of Bayesian network structure learning by constraining the search space. These constraints can come in the form of ancestral constraints that relate to the existence of paths between nodes. When the constraints are compiled into a directed acyclic graph, the complexity of learning with ancestral constraints is connected to the number of ideals of the constraint graph. First, we consider precedence constraints which define a partial order that the structure must obey. Taking the path cover number of the constraint graph as a parameter, we extend earlier results to the problems of sampling and weighted counting of network structures. We also consider the problems with related ancestral constraints which state that a node must or cannot be an ancestor of another. With positive ancestral constraints, we show that the problems are tractable under the additional assumption that the constraint graph has only a small number of incomparable edges. On the other hand, the optimization problem is NP-hard with negative ancestral constraints when the path cover number is at least two. Finally, we show that these problems become fixed-parameter tractable if the constraints are compatible with a subclass of partial orders called bucket orders.