When you come at the kings you best not miss

TL;DR

This paper introduces a new strategy for identifying a nearly-king vertex in a tournament with fewer queries, improving the known bounds by leveraging a novel structural result.

Contribution

It presents a strategy that finds a -king using O(n^{4/3} polylog n) queries, improving previous bounds and introducing a new structural theorem for tournaments.

Findings

Achieves -king identification with fewer queries.

Provides a new structural result for tournaments.

Improves upon previous query complexity bounds.

Abstract

A tournament is an orientation of a complete graph. We say that a vertex $x$ in a tournament $\vec T$ controls another vertex $y$ if there exists a directed path of length at most two from $x$ to $y$. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu (SIAM J. Comput., 2003) in investigating the query complexity of finding a king, that is, the number of arcs in $\vec T$ one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always has to query at least $\Omega(n^{4/3})$ arcs and provided a strategy that queries at most $O(n^{3/2})$. While this upper bound has not yet been improved for the original problem, Biswas et al. (Frontiers in Algorithmics, 2017) proved that with $O(n^{4/3})$ queries one can identify a semi-king, meaning a vertex which controls at least half of all vertices. Our contribution is a novel strategy which improves upon the number of controlled vertices: using $O(n^{4/3} \operatorname{polylog} n)$ queries, we can identify a $(\frac{1}{2}+\frac{2}{17})$-king. To achieve this goal we use a novel structural result for tournaments.