Hardness of Finding Kings and Strong Kings

TL;DR

The paper investigates the query complexity of finding kings in directed graphs and strong kings in tournaments. It proves that both the randomized and deterministic query complexities for deciding the existence of a king in general digraphs are $\Theta(n^2)$, and that the randomized/ deterministic complexities for finding a strong king in tournaments are also $\Theta(n^2)$. The authors introduce two balanced-tournament constructions, $\Delta_n$ and $U_n$, and use them to derive hard instances and analyze path-count properties. This work clarifies the limits of subquadratic queries for these problems and proposes constructions that may inform further study of tournament solutions and related notions.

Abstract

A king in a directed graph is a vertex $v$ such that every other vertex is reachable from $v$ via a path of length at most $2$. It is well known that every tournament (a complete graph where each edge has a direction) has at least one king. Our contributions in this work are: - We show that the query complexity of determining existence of a king in arbitrary $n$-vertex digraphs is $Θ(n^2)$. This is in stark contrast to the case where the input is a tournament, where Shen, Sheng, and Wu [SICOMP'03] showed that a king can be found in $O(n^{3/2})$ queries. - In an attempt to increase the "fairness" in the definition of tournament winners, Ho and Chang [IPL'03] defined a strong king to be a king $k$ such that, for every $v$ that dominates $k$, the number of length-$2$ paths from $k$ to $v$ is strictly larger than the number of length-$2$ paths from $v$ to $k$. We show that the query complexity of finding a strong king in a tournament is $Θ(n^2)$. This answers a question of Biswas, Jayapaul, Raman, and Satti [DAM'22] in the negative. A key component in our proofs is the design of specific tournaments where every vertex is a king, and analyzing certain properties of these tournaments. We feel these constructions and properties are independently interesting and may lead to more interesting results about tournament solutions.

Figures (3)

• Figure 1: Illustration of Construction \ref{['con:triangle']}.
• Figure 2: The tournament $U_5$ and the tournament $\sigma(U_5, 3) \in \mathrm{Aut}(U_5)$ obtained by rotating the labels cyclically clockwise by 3 positions.
• Figure 3: Construction of $C$ from the proof of Theorem \ref{['thm:randomised-king-existence']}. $C[A]$ is a balanced tournament, and $C[A']$ is a relabelling of $\Delta_n$ from Construction \ref{['con:triangle']}

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Tournament, Subgraph, Subtournament
• Definition 2.2: King
• Lemma 2.3: M80
• Lemma 2.4: HC03
• proof
• Lemma 2.5: Yao's Minimax Principle
• Lemma 3.2
• proof