{
  "abstract": "Lee and Sidford [LS19] showed that the linear program $\\min\\{c^\\top x\\ :\\ A^\\top x=b,\\ l\\leq x\\leq u\\}$ over $x\\in\\mathbb{R}^m$, where $A\\in\\mathbb{R}^{m\\times n}$, can be solved to accuracy $ε$ using $O(\\sqrt n\\log^{13}m\\cdot\\log(mU/ε))$ solves of linear systems in $A^\\top\\mathbf DA$ for positive diagonal matrices $\\mathbf D$, where $U$ bounds the magnitudes of the input and of the initial point. Theirs is the first such bound governed by $\\mathrm{rank}(A)=n$ rather than by the number of constraints $m$. Song [Son19] mentioned that reducing the $\\log^{13}m$ factor is an interesting future direction. Given an interior point and a finite certified magnitude bound $U$, we give an algorithm that uses $O(\\sqrt n\\log^{4}m\\cdot\\log(mU/ε))$ solves of such linear systems, improving the Lee--Sidford bound by a factor of $\\log^{9}m$. We conjecture that $O(\\sqrt n\\log(mU/ε))$ such linear-system solves suffice.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30853",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhao Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/829wk0m3c1e4j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30853",
  "fulltext_url": null,
  "id": "829wk0m3c1e4j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30853",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7214657180",
  "pdf_url": "https://arxiv.org/pdf/2609.30853",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/829wk0m3c1e4j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30853",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30853"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Faster Linear Programming with $\\sqrt{\\mathrm{rank}}$ Linear System Solves",
  "title_text": "Faster Linear Programming with √(rank) Linear System Solves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/829wk0m3c1e4j",
  "venue": "arXiv",
  "versions": []
}