{
  "abstract": "We give an efficient algorithm with improved algorithmic guarantees for the (offline) Beck--Fiala problem when the sets have bounded size. Let $A$ be an arbitrary matrix $A\\in\\{0,1\\}^{m\\times n}$ with at most $d$ ones per column and at most $s$ ones per row. Let $\\log^*$ denote the iterated logarithm and $\\ell_j$ denote the $j$-fold composition of log. Assume $s\\le\\exp(O(\\sqrt d))$. We provide an efficient algorithm that, for arbitrary sparsity $d$, gives $O(\\sqrt d(1+\\log^*n))$ discrepancy. Moreover, if $d\\ge\\ell_j(n)$ for a fixed integer $j\\ge1$, the algorithm gives $O_j(\\sqrt d)$ discrepancy. The proof is a bootstrapping scheme using the Bansal-Jiang algorithm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19714",
  "author_total": 1,
  "authors": [
    {
      "name": "Dylan J. Altschuler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/66p0s3hz6e2r2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19714",
  "fulltext_url": null,
  "id": "66p0s3hz6e2r2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19714",
  "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.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.19714",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/66p0s3hz6e2r2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19714",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19714"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Improved Algorithms for Beck--Fiala with Bounded Sets",
  "title_text": "Improved Algorithms for Beck--Fiala with Bounded Sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/66p0s3hz6e2r2",
  "venue": "arXiv",
  "versions": []
}