  Inferring probabilistic networks from data is a notoriously difficult task.
Under various goodness-of-fit measures, finding an optimal network is NP-hard,
even if restricted to polytrees of bounded in-degree. Polynomial-time
algorithms are known only for rare special cases, perhaps most notably for
branchings, that is, polytrees in which the in-degree of every node is at most
one. Here, we study the complexity of finding an optimal polytree that can be
turned into a branching by deleting some number of arcs or nodes, treated as a
parameter.
  We show that the problem can be solved via a matroid intersection formulation
in polynomial time if the number of deleted arcs is bounded by a constant. The
order of the polynomial time bound depends on this constant, hence the
algorithm does not establish fixed-parameter tractability when parameterized by
the number of deleted arcs. We show that a restricted version of the problem
allows fixed-parameter tractability and hence scales well with the parameter.
We contrast this positive result by showing that if we parameterize by the
number of deleted nodes, a somewhat more powerful parameter, the problem is not
fixed-parameter tractable, subject to a complexity-theoretic assumption.
