The FEEDBACK VERTEX SET problem, given an undirected graph G and an integer
k, asks whether there exists a set of at most k vertices that hits all cycles
in the graph G. FEEDBACK VERTEX SET was intensively studied from the
parameterized perspective. In this paper we consider the SUBSET FEEDBACK VERTEX
SET problem, where an instance comes additionally with a subset of vertices S
and we ask for a set of at most k vertices that hits all simple cycles passing
through S. The parameterized complexity of SUBSET FEEDBACK VERTEX SET was
posted as an open problem independently by K. Kawarabayashi and by S. Saurabh
in 2009. We begin by showing that this problem is FPT when parameterized by
|S|. Moreover, we present an algorithm which reduces the size of |S| to O(k^3)
in 2^O(k log k)n^O(1) time using kernelization techniques such as the
2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two results
prove that the SUBSET FEEDBACK VERTEX SET problem is FPT when parameterized by
k and the time complexity of our algorithm is 2^O(k log k)n^O(1).

The SUBSET FEEDBACK VERTEX SET problem is closely related to terminal
separation problems such as MULTIWAY CUT and MULTICUT: SUBSET FEEDBACK VERTEX
SET is a generalisation of MULTIWAY CUT and with an additional constraint
SUBSET FEEDBACK VERTEX SET becomes as hard as MULTICUT. It is known that
MULTIWAY CUT is FPT when parameterized by the solution size k but for MULTICUT
the existence of such a FPT algorithm is a major and long-standing open problem
in the FPT field. In some sense SUBSET-FVS falls between MULTIWAY CUT and
MULTICUT. Since SUBSET-FVS is at least as hard as NODE MULTIWAY CUT and it is
still FPT, we make a step towards solving the MULTICUT problem.