An approximation algorithm or feedback vertex sets in tournaments

We obtain a necessary and sufficient condition in terms of forbidden struct ures for tournaments to possess the min-max relation on packing and coverin g directed cycles, together with strongly polynomial time algorithms for th e feedback vertex set problem and the cycle packing problem in this class o f tournaments. Applying the local ratio technique of Bar-Yehuda and Even to the forbidden structures, we nd a 2.5-approximation polynomial time algori thm for the feedback vertex set problem in any tournament.