In knockout tournaments, players compete in successive rounds, with losers eliminated and winners advancing until a single champion remains. Given a tournament digraph $D$, which encodes the outcomes of all possible matches, and a designated player $v^* \in V(D)$, the \textsc{Tournament Fixing} problem (TFP) asks whether the tournament can be scheduled in a way that guarantees $v^*$ emerges as the winner. TFP is known to be NP-hard, but is fixed-parameter tractable (FPT) when parameterized by structural measures such as the feedback arc set (fas) or feedback vertex set (fvs) number of the tournament digraph. In this paper, we introduce and study two new structural parameters: the number of players who can defeat $v^*$ (i.e., the in-degree of $v^*$, denoted by $k$) and the number of players that $v^*$ can defeat (i.e., the out-degree of $v^*$, denoted by $\ell$). A natural question is that: can TFP be efficiently solved when $k$ or $\ell$ is small? We answer this question affirmatively by showing that TFP is FPT when parameterized by either the in-degree or out-degree of $v^*$. Our algorithm for the in-degree parameterization is particularly involved and technically intricate. Notably, the in-degree $k$ can remain small even when other structural parameters, such as fas or fvs, are large. Hence, our results offer a new perspective and significantly broaden the parameterized algorithmic understanding of the \textsc{Tournament Fixing} problem.

翻译：在淘汰制锦标赛中，选手们逐轮进行比赛，败者淘汰，胜者晋级，直至产生唯一的冠军。给定一个编码所有可能比赛结果的锦标赛有向图$D$，以及一个指定选手$v^* \in V(D)$，\textsc{Tournament Fixing}问题（TFP）询问是否能够通过安排赛程来确保$v^*$成为冠军。已知TFP是NP难的，但当以锦标赛有向图的结构性度量（如反馈弧集（fas）数或反馈顶点集（fvs）数）作为参数时，该问题是固定参数可解（FPT）的。本文引入并研究了两个新的结构参数：能够击败$v^*$的选手数量（即$v^*$的入度，记为$k$），以及$v^*$能够击败的选手数量（即$v^*$的出度，记为$\ell$）。一个自然的问题是：当$k$或$\ell$较小时，TFP能否被高效求解？我们通过证明TFP在以$v^*$的入度或出度为参数时是FPT的，从而对这个问题给出了肯定的回答。我们针对入度参数化的算法尤为复杂且技术性强。值得注意的是，即使其他结构参数（如fas或fvs）很大，入度$k$仍可能保持较小。因此，我们的研究结果为\textsc{Tournament Fixing}问题的参数化算法理解提供了新的视角并显著拓宽了其范围。