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

TL;DR

This paper investigates the computational complexity of ensuring a specific player wins in knockout tournaments by introducing new parameters related to the player's potential opponents and demonstrating fixed-parameter tractability under these parameters.

Contribution

The paper introduces and analyzes two new structural parameters, the in-degree and out-degree of the designated player, showing fixed-parameter tractability for the tournament fixing problem when parameterized by these measures.

Findings

TFP is fixed-parameter tractable when parameterized by the in-degree of the player.

TFP is fixed-parameter tractable when parameterized by the out-degree of the player.

The in-degree parameter can be small even if other structural parameters are large.

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.