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

TL;DR

The paper addresses the Tournament Fixing problem (TFP): given a tournament $D$ and a favorite $v^*$, does a seeding exist that guarantees $v^*$ wins? It introduces two local parameters, $k=|N_{ ext{in}}(v^*)|$ and $oldsymbol{\\ell}=|N_{ ext{out}}(v^*)|$, and proves TFP is fixed-parameter tractable (FPT) with respect to either parameter. The out-degree case yields a straightforward $2^{2^{oldsymbol{\\ell}}} ext{--time}$ FPT algorithm with ETH-based lower bounds, while the in-degree case requires deeper structural insight via winning-witness forests (WWFs) and a color-coding reduction to a restricted Subgraph Isomorphism problem, yielding an FPT algorithm with a carefully analyzed running time. The work broadens the parameterized algorithmic toolbox for TFP by offering a new player-centric lens and highlighting open questions about related subset fas/fvs parameters. Together, these results enhance understanding of when local neighborhood structure around the favorite player governs tractability in tournament manipulation problems.

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.

Figures (4)

• Figure 1: An illustration of the hierarchy of the six 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 $\FPT$ (including our results), while the orange region (lower section) indicates unresolved cases.
• Figure 2: $T_u$ is a UBA with 16 vertices, and the subtree $T'$ formed by $T_{v_1}$, $T_{v_2}$ and $v$ is a UBA with $2^2 = 4$ vertices.
• Figure 3: The union of matches among players from $W, X \setminus W$ and a new combined tournament
• Figure 4: The construction of $F'$ and $D'$.

Theorems & Definitions (24)

• Definition 1: match set and sequence
• Definition 2: unlabeled binomial arborescence
• Proposition 1: williams2010fixing
• Lemma 1: gupta2018winning
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• Definition 3: nice rounds and nice seedings