  Optimal Morse matchings reveal essential structures of cell complexes which
lead to powerful tools to study discrete geometrical objects, in particular
discrete 3-manifolds. However, such matchings are known to be NP-hard to
compute on 3-manifolds, through a reduction to the erasability problem.
  Here, we refine the study of the complexity of problems related to discrete
Morse theory in terms of parameterized complexity. On the one hand we prove
that the erasability problem is W[P]-complete on the natural parameter. On the
other hand we propose an algorithm for computing optimal Morse matchings on
triangulations of 3-manifolds which is fixed-parameter tractable in the
treewidth of the bipartite graph representing the adjacency of the 1- and
2-simplexes. This algorithm also shows fixed parameter tractability for
problems such as erasability and maximum alternating cycle-free matching. We
further show that these results are also true when the treewidth of the dual
graph of the triangulated 3-manifold is bounded. Finally, we investigate the
respective treewidths of simplicial and generalized triangulations of
3-manifolds.
