Learning the structure of a Bayesian network from data is one of the key problems in probabilistic graphical models. Unfortunately, the problem is NP-hard and this has motivated recent works where the problem has been studied from the perspective of algorithmic paradigms meant for coping with hardness, such as parameterized complexity. We contribute to this area by designing fixed parameter tractable algorithms (FPT) to learn the Bayesian network structure when only a few variables are important. In particular, we study score-based structure learning where each graph is given with a score, based on how well it fits to the data, and the goal is to select the acyclic directed graph (DAG) that maximizes the score. Typically, one uses decomposable scores, where the score of a DAG is the sum of local scores for node-parent set pairs. We study a variant of this problem in which our objective is to find a k-heavy DAG, which is a DAG whose k most scoring nodes have a total score of at least some target value ℓ. We show that

1. if there is a k-heavy DAG with a maximum degree of d, then we can learn it in time f(k,d)n^O(d) and

2. if there is a k-heavy DAG whose moralized graph has a treewidth of t and a maximum degree of t, then we can learn it in time f(k,t)n^O(t).

These algorithms leverage the color-coding technique from the field of Parameterized Complexity in a non-trivial manner.