  In this paper we present the first dynamic algorithms for the problem of
Feedback Arc Set in Tournaments (FAST) and the problem of Feedback Vertex Set
in Tournaments (FVST). Our algorithms maintain a dynamic tournament on n
vertices altered by redirecting the arcs, and answer if the tournament admits a
feedback arc set (or respectively feedback vertex set) of size at most K, for
some chosen parameter K. For dynamic FAST we offer two algorithms. In the
promise model, where we are guaranteed, that the size of the solution does not
exceed g(K) for some computable function g, we give an $O(\sqrt{g(K)})$ update
and $O(3^K K \sqrt{K})$ query algorithm. In the general setting without any
promise, we offer an $O(\log^2 n)$ update and $O(3^K K \log^2 n)$ query time
algorithm for dynamic FAST. For dynamic FVST we offer an algorithm working in
the promise model, which admits $O(g^5(K))$ update and $O(3^K K^3 g(K))$ query
time.
