From Donkeys to Kings in Tournaments

TL;DR

The paper investigates the complexity of finding $k$ kings in a tournament under both edge-query (sublinear) and RAM (full-input) models. It identifies a phase transition: for $k \le 3$ there are sublinear randomized query algorithms (and near-linear deterministic ones), while for $k \ge 4$ randomized query complexity is $\Omega(n^2)$, indicating a sharp increase in difficulty. In the RAM model, it provides an $O(kn^{2})$ deterministic algorithm (optimal for fixed $k$) and an $O(n^{\omega})$ algorithm for large $k$, where $\omega \approx 2.3715$; the bounds are connected via a fine-grained reduction from a variant of triangle detection. The work uses the Yao Minimax Principle and pivot-based strategies to derive upper and lower bounds, and it gives conditional optimality results by linking $k$-king complexity to the $\exists \forall$-Triangle Problem. These results deepen our understanding of simple social-choices-like structures in tournaments and illuminate how fine-grained complexity and linear-algebraic techniques interact in graph-theoretic problems.

Abstract

A tournament is an orientation of a complete graph. A vertex that can reach every other vertex within two steps is called a \emph{king}. We study the complexity of finding $k$ kings in a tournament graph. We show that the randomized query complexity of finding $k \le 3$ kings is $O(n)$, and for the deterministic case it takes the same amount of queries (up to a constant) as finding a single king (the best known deterministic algorithm makes $O(n^{3/2})$ queries). On the other hand, we show that finding $k \ge 4$ kings requires $Ω(n^2)$ queries, even in the randomized case. We consider the RAM model for $k \geq 4$. We show an algorithm that finds $k$ kings in time $O(kn^2)$, which is optimal for constant values of $k$. Alternatively, one can also find $k \ge 4$ kings in time $n^ω$ (the time for matrix multiplication). We provide evidence that this is optimal for large $k$ by suggesting a fine-grained reduction from a variant of the triangle detection problem.

Figures (3)

• Figure 1: If there is no edge between $A,B,C$ to $k_1, k_2, k_3$ then that means that the edges goes from $k_i$ to the set.
• Figure 2: Relationship between $A$, $A'$$B$ and $C$. A red arrow indicates that this is the direction of the edge if the two vertices have an edge in $G$. A blue arrow indicates that this is the direction of the edge if there is no edge in $G$. A black arrow indicates that this is the direction of the edge regardless of $G$. For $A$ and $A'$, we have $a_i \to a'_j$ if $i = j$ and otherwise $a'_j \to a_i$.
• Figure 3: Relationship between $x_1$, $x_2$ and the remaining sets.

Theorems & Definitions (48)

• Definition 1: King
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Informal
• Definition 2
• Definition 3
• Definition 4
• Definition 5: King