{
  "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].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03838",
  "author_total": 3,
  "authors": [
    {
      "name": "Ziad Ismaili Alaoui"
    },
    {
      "name": "Oded Lachish"
    },
    {
      "name": "Nikhil S. Mande"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f31v0gp778faz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03838",
  "fulltext_url": null,
  "id": "f31v0gp778faz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03838",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03838",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f31v0gp778faz.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.03838",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03838"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A King for Almost Everyone",
  "title_text": "A King for Almost Everyone",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/f31v0gp778faz",
  "venue": "arXiv",
  "versions": []
}