{
  "abstract": "Let $Δ_u(d,n)$ be the maximum graph diameter of a pointed $d$-dimensional polyhedron with $n$ facets. Using a curved-path encoding of the iterated Kalai--Kleitman recurrence, we prove, uniformly over $n\\geq d\\geq 4$, \\[ Δ_u(d,n) \\leq (n-d)^{\\log_2 G_d},\\qquad G_d = (4\\ln 2+o(1))\\frac{d}{(\\ln d)^2}, \\] where the asymptotic expression for $G_d$ is understood as $d\\to\\infty$, improving the exponent of the previous best quasi-polynomial bound by an additional logarithmic factor. With the quantitative $d$-step reduction, we also obtain the complementary excess-based bound \\[ Δ_u(d,n) \\leq (n-d)^{\\frac{1}{2}\\log_2(n-d)+O(1)}, \\] where the implied constant is absolute. In the regime $n - d = Θ(d)$, the latter bound is asymptotically stronger and halves the leading coefficient in the exponent. We further examine their behavior as $n$ grows relative to $d$, obtaining sharper exponents when $n = d^{1/γ+o(1)}$ for fixed $0<γ<1$ and an almost-linear bound in the deep-tail regime $(\\ln n)/d\\to\\infty$. We also show that the leading term of the general bound is sharp within this positive path-counting framework.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33434",
  "author_total": 3,
  "authors": [
    {
      "name": "Tianhao Liu"
    },
    {
      "name": "Dongdong Ge"
    },
    {
      "name": "Yinyu Ye"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/19mwx8h48f7gk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33434",
  "fulltext_url": null,
  "id": "19mwx8h48f7gk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33434",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214740186",
  "pdf_url": "https://arxiv.org/pdf/2609.33434",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/19mwx8h48f7gk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33434",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33434"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Curved-Path Refinement of the Kalai--Kleitman Diameter Bound",
  "title_text": "A Curved-Path Refinement of the Kalai--Kleitman Diameter Bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/19mwx8h48f7gk",
  "venue": "arXiv",
  "versions": []
}