Fixed-parameter tractability for the subset feedback set problem and the S-cycle packing problem

We investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our problem setting is that we are given a graph and a vertex set S called “terminals”. Our purpose here is to consider the following problems: 1. The feedback set problem with respect to the terminals S. We call it the subset feedback set problem. 2. The cycle packing problem with respect to the terminals S, i.e., each cycle has to contain a vertex in S (such a cycle is called an S-cycle). We call it the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely; 1. For fixed k, we can either find a vertex set X of size k such that G−X has no S-cycle, or conclude that such a vertex set does not exist in O(nm) time, where n is the number of vertices of the input graph and m is the number of edges of the input graph. 2. For fixed k, we can either find k vertex-disjoint S-cycles or conclude that such k disjoint cycles do not exist in O(n) time.