The Complexity of Tournament Fixing: Subset FAS Number and Acyclic Neighborhoods

Abstract

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.