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

TL;DR

This paper proves that TFP stays NP-hard even when the subset FAS number of v^* is constant and either the subgraph induced by the in-neighbors or the out-neighbors is acyclic, and shows that TFP becomes FPT parameterized by the subset FAS number of v^*.

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.

Figures (5)

• Figure 1: An illustration of the hierarchy of the eight parameters, where an arc from parameter $x$ to parameter $y$ denotes $x \leq y$. The green region (upper section) marks parameters for which TFP is proven $\FPT$, while the red region (lower section) indicates para-$\NP$-hard cases (our results).
• Figure 2: An illustration for a special instance $(D, v^*)$.
• Figure 3: An illustration for the complete binary tree $T$ corresponding to a winner seeding $\sigma$ for a special instance $(D, v^*)$, where $a'$ is a player in $N_{\mathrm{out}}(v^*)$.
• Figure 4: An illustration for the constructed tournament $D'$.
• Figure 5: An illustration for a no-insatnce $(D, v^*)$ where $v^*$ is a king and $\forall b\in N_{\mathrm{in}}(v^*), out(b) \leq 2out(v^*)$.

Theorems & Definitions (24)

• definition 1: match set and sequence
• definition 2
• lemma 1
• proof
• theorem 1
• proof
• corollary 1
• corollary 2
• theorem 2
• proof