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

2026-02-13

AI Summary

This study investigates the computational complexity of the tournament fixing problem (TFP) parameterized by the subset feedback arc set (subset FAS) number. By integrating parameterized complexity analysis, feedback arc set theory, and structural decomposition of directed graphs, the authors establish that TFP remains NP-hard even when the subset FAS number is constant, provided that either the in-neighborhood or out-neighborhood subgraph is acyclic. However, when both in- and out-neighborhood subgraphs are acyclic, the problem becomes fixed-parameter tractable (FPT). This result precisely delineates the tractability boundary of TFP under this structural parameter and yields a sufficient condition for a player to guarantee victory, thereby clarifying the theoretical limits of the subset FAS number as a parameter for capturing problem structure.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Tournament Fixing Problem
• Subset FAS Number
• Fixed-Parameter Tractability
• Acyclic Neighborhoods
• NP-hardness

Innovation

• Tournament Fixing Problem
• Subset Feedback Arc Set
• Fixed-Parameter Tractability
• Acyclic Neighborhoods
• NP-hardness