How Hard Is It to Rig a Tournament When Few Players Can Beat or Be Beaten by the Favorite?

2026-01-13

AI Summary

This study addresses the Tournament Fixing Problem, which asks whether a given player can be made to win through an appropriate arrangement of the tournament bracket. The work introduces, for the first time, structural parameters based on the in-degree (the number of players who can defeat the given player) and out-degree (the number of players the given player can defeat) of the target player, and develops fixed-parameter tractable (FPT) algorithms with respect to these parameters. The results show that when either the in-degree or out-degree of the designated player is bounded by a constant, the problem becomes solvable in polynomial time. This substantially broadens the range of tractable instances, particularly in scenarios where traditional parameters are large but the newly introduced ones remain small, thereby deepening the understanding of the parameterized complexity landscape of the Tournament Fixing Problem.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Search and Optimization: Algorithm Configuration
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Tournament Fixing
• knockout tournaments
• parameterized complexity
• in-degree
• out-degree

Innovation

• Tournament Fixing
• fixed-parameter tractability
• in-degree parameterization
• out-degree parameterization
• knockout tournaments