Learning Bayesian Networks Under Sparsity Constraints: A Parameterized Complexity Analysis
share
We study the problem of learning the structure of an optimal Bayesian network D when additional constraints are posed on the DAG D or on its moralized graph. More precisely, we consider the constraint that the moralized graph can be transformed to a graph from a sparse graph class Π by at most k vertex deletions. We show that for Π being the graphs with maximum degree 1, an optimal network can be computed in polynomial time when k is constant, extending previous work that gave an algorithm with such a running time for Π being the class of edgeless graphs [Korhonen Parviainen, NIPS 2015]. We then show that further extensions or improvements are presumably impossible. For example, we show that when Π is the set of graphs with maximum degree 2 or when Π is the set of graphs in which each component has size at most three, then learning an optimal network is NP-hard even if k=0. Finally, we show that learning an optimal network with at most k edges in the moralized graph presumably has no f(k)· |I|^𝒪(1)-time algorithm and that, in contrast, an optimal network with at most k arcs in the DAG D can be computed in 2^𝒪(k)· |I|^𝒪(1) time where |I| is the total input size.
READ FULL TEXT
