Given a tournament $T$ where $S_1$ and $S_2$ be two acyclic sub-tournament of $T$.
Is the following problem NP-Complete: Finding a maximum acyclic sub-tournament $S$, which is subset of $S_1 \cup S_2$?
Can the given problem be solved in polynomial time? If not please state the NP-Completeness.
By keeping $S_1$ as such and removing only the vertices from $S_2$, an $S'$ maximal acyclic tournament belonging to $S_1 \cup S_2$ can be obtained in polynomial time. The solution $S'$ thus obtained may not be the same as the maximum acyclic sub-tournament $S$.
The polynomial time algorithm is based on compression step in iterative compression algorithm for feedback vertex set in tournament from the paper
Fixed-parameter tractability results for feedback set problems in tournaments, Michael Dom, Jiong Guo, Falk Hüffner, Rolf Niedermeier, Anke Truss, Journal of Discrete Algorithms 8 (2010) 76–86.
If finding a maximal acyclic sub-tournament $S$ is NP-complete then I have no choice except to find $S'$, so I wish to know whether finding $S$ is NP-complete or not.