  A tournament is a directed graph T such that every pair of vertices are
connected by an arc. A feedback vertex set is a set S of vertices in T such
that T - S is acyclic. In this article we consider the Feedback Vertex Set
problem in tournaments. Here input is a tournament T and integer k, and the
task is to determine whether T has a feedback vertex set of size at most k. We
give a new algorithm for Feedback Vertex Set in Tournaments. The running time
of our algorithm is upper bounded by O(1.618^k + n^{O(1)}) and by O(1.46^n).
Thus our algorithm simultaneously improves over the fastest known parameterized
algorithm for the problem by Dom et al. running in time O(2^kk^{O(1)} +
n^{O(1)}), and the fastest known exact exponential time algorithm by Gaspers
and Mnich with running time O(1.674^n).
  On the way to prove our main result we prove a new partitioning theorem for
undirected graphs. In particular we show that the vertices of any undirected
m-edge graph of maximum degree d can be colored white or black in such a way
that for each of the two colors, the number of edges with both endpoints of
that color is between m/4-d/2 and m/4+d/4.
