Polynomial kernels for deletion to classes of acyclic digraphs

We consider the problem to find a set X of vertices (or arcs) with |X|≤k in a given digraph G such that D=G−X is an acyclic digraph. In its generality, this is Directed Feedback Vertex Set (or Directed Feedback Arc Set); the existence of a polynomial kernel for these problems is a notorious open problem in the field of kernelization, and little progress has been made. In this paper, we consider the vertex deletion problem with an additional restriction on D, namely that D must be an out-forest, an out-tree, or a (directed) pumpkin. Our main results show that for each of the above three restrictions we can obtain a kernel with kO(1) vertices on general digraphs. We also show that the arc deletion problem with the first two restrictions (out-forest and out-tree) can be solved in polynomial time; this contrasts the status of the vertex deletion problem of these restrictions, which is NP-hard even for acyclic digraphs. The arc and vertex deletion problem with the third restriction (pumpkin), however, can be solved in polynomial time for acyclic digraphs, but becomes NP-hard for general digraphs. We believe that the idea of restricting D could yield new insights towards resolving the status of Directed Feedback Vertex Set.