Hardness of Finding Kings and Strong Kings

2025-04-27

AI Summary

This paper investigates the query complexity of identifying “kings” (vertices that reach all others within distance two) in arbitrary directed graphs and “strong kings” (vertices strictly dominating all vertices they dominate in the number of length-two paths) in tournaments. Using combinatorial game-theoretic analysis, extremal graph constructions, and structural modeling of tournaments, we establish tight Θ(n²) query lower bounds for both problems—resolving open questions in decision complexity. Specifically, we prove that determining whether a king exists in an arbitrary directed graph requires Θ(n²) adjacency queries, refuting the possibility of subquadratic algorithms; likewise, finding a strong king in a tournament also admits a Θ(n²) tight lower bound, disproving prior conjectures on efficient computation. Furthermore, we construct a novel family of tournaments where every vertex is a king, revealing deep structural and fairness properties. These results precisely characterize the query complexity of both problems and broaden the interface between tournament theory and computational decision complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Adversarial Search
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Models for Web evolution

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 $Theta(n^2)$. 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 $Theta(n^2)$. 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.

Problem

• Determine query complexity for finding kings in digraphs
• Investigate query complexity for strong kings in tournaments
• Analyze properties of tournaments where every vertex is a king

Innovation

• Determining king existence requires Θ(n²) queries
• Finding strong kings needs Θ(n²) query complexity
• Constructs specific tournaments where all vertices are kings