  The goal of this paper is to open up a new research direction aimed at
understanding the power of preprocessing in speeding up algorithms that solve
NP-hard problems exactly. We explore this direction for the classic Feedback
Vertex Set problem on undirected graphs, leading to a new type of graph
structure called antler decomposition, which identifies vertices that belong to
an optimal solution. It is an analogue of the celebrated crown decomposition
which has been used for Vertex Cover. We develop the graph structure theory
around such decompositions and develop fixed-parameter tractable algorithms to
find them, parameterized by the number of vertices for which they witness
presence in an optimal solution. This reduces the search space of
fixed-parameter tractable algorithms parameterized by the solution size that
solve Feedback Vertex Set.
