The \textsc{Tournament Fixing Problem} (TFP) asks whether a knockout tournament can be scheduled to guarantee that a given player $v^*$ wins. Although TFP is NP-hard in general, it is known to be \emph{fixed-parameter tractable} (FPT) when parameterized by the feedback arc/vertex set number, or the in/out-degree of $v^*$ (AAAI 17; IJCAI 18; AAAI 23; AAAI 26). However, it remained open whether TFP is FPT with respect to the \emph{subset FAS number of $v^*$} -- the minimum number of arcs intersecting all cycles containing $v^*$ -- a parameter that is never larger than the aforementioned ones (AAAI 26). In this paper, we resolve this question negatively by proving that TFP stays NP-hard even when the subset FAS number of $v^*$ is constant $\geq 1$ and either the subgraph induced by the in-neighbors $D[N_{\mathrm{in}}(v^*)]$ or the out-neighbors $D[N_{\mathrm{out}}(v^*)]$ is acyclic. Conversely, when both $D[N_{\mathrm{in}}(v^*)]$ and $D[N_{\mathrm{out}}(v^*)]$ are acyclic, we show that TFP becomes FPT parameterized by the subset FAS number of $v^*$. Furthermore, we provide sufficient conditions under which $v^*$ can win even when this parameter is unbounded.

翻译：\textsc{锦标赛操控问题}（TFP）探讨的是能否通过安排淘汰赛赛程来确保特定选手$v^*$获胜。尽管TFP在一般情况下是NP难的，但已知当以反馈弧/顶点集数或$v^*$的入度/出度为参数时，该问题是\emph{固定参数可解}的（AAAI 17; IJCAI 18; AAAI 23; AAAI 26）。然而，对于以\emph{$v^*$的子集FAS数}——即与所有包含$v^*$的环相交的最小弧数（该参数始终不大于前述参数）——为参数时TFP是否仍为固定参数可解的问题，此前一直悬而未决（AAAI 26）。本文通过证明即使$v^*$的子集FAS数为常数$\geq 1$，且入邻域诱导子图$D[N_{\mathrm{in}}(v^*)]$或出邻域诱导子图$D[N_{\mathrm{out}}(v^*)]$之一为无环图时，TFP仍保持NP难，从而对该问题给出了否定解答。反之，当$D[N_{\mathrm{in}}(v^*)]$与$D[N_{\mathrm{out}}(v^*)]$均为无环图时，我们证明TFP在以$v^*$的子集FAS数为参数时具有固定参数可解性。此外，我们还给出了在该参数无界情况下$v^*$仍可能获胜的充分条件。