Improved bounds for minimal feedback vertex sets in tournaments.

We study feedback vertex sets (FVS) in tournaments, which are orientations of complete graphs. As our main result, we show that any tournament on n nodes has at most 1.5949 n minimal FVS. This significantly improves the previously best upper bound of 1.6667 n by Fomin et al. [STOC 2016] and 1.6740 n by Gaspers and Mnich [J. Graph Theory72(1):72-89, 2013]. Our new upper bound almost matches the best-known lower bound of 21n/7≈1.5448n, due to Gaspers and Mnich. Our proof is algorithmic, and shows that all minimal FVS of tournaments can be enumerated in time O(1.5949n).