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arXiv · 2611.03838

A King for Almost Everyone

Ziad Ismaili Alaoui·Oded Lachish·Nikhil S. Mande
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Abstract

A tournament is an orientation of a complete graph. A vertex $v$ controls a vertex $u\neq v$ if there is a directed path of length at most two from $v$ to $u$, and a $δ$-king is one that controls at least $δ(n-1)$ vertices. In the query complexity model, each query reveals the orientation of an edge. Shen, Sheng and Wu [SICOMP'03] established an $Ω(n^{4/3})$ lower bound on the deterministic query complexity of finding a $1$-king, and a non-matching upper bound of $O(n^{3/2})$. Neither bound has been improved in over two decades, despite substantial progress on variants of the problem. For $δ$-kings with $δ< 1$, Biswas, Jayapaul, Raman and Satti [Discrete Applied Mathematics'22] showed that $O(n^{4/3})$ queries suffice to find a $\frac12$-king. Lachish, Reidl and Trehan [FSTTCS'22] improved this guarantee to $\frac12+\frac{2}{17}$ using $\widetilde{O}(n^{4/3})$ queries, and asked whether a larger fraction could be achieved within the same query bound. We answer this question affirmatively for every fixed fraction below one: for every fixed $\varepsilon\in(0,1)$, we give a deterministic algorithm that finds a $(1-\varepsilon)$-king using $O(n^{4/3})$ queries. The design of our algorithm is informed by a structural analysis combining a sparse regularity lemma due to Allen, Kohayakawa, Mota and Parente [Electronic Journal of Combinatorics'14] with a new sampling argument based on a reachability theorem of Afek, Gafni and Linial [arXiv'19].

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BibTeXRIS

Ziad Ismaili Alaoui, Oded Lachish, Nikhil S. Mande. 2026-11-02. A King for Almost Everyone. https://arxiv.org/abs/2611.03838

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