{
  "abstract": "Let $M$ be a matroid, let $\\ell:E(M)\\to G$ be a labeling of its elements by an abelian group, and let $F\\subseteq G$ be finite. A base is $F$-avoiding if the label sum of its elements is not in $F$. We prove that, for arbitrary costs, a min-cost $F$-avoiding base can be obtained from any min-cost base using at most $|F|$ exchanges. This resolves the weighted proximity conjecture by Hörsch, Imolay, Mizutani, Oki, and Schwarcz. Our main technical tool is to replace the assumption of Schrijver--Seymour Conjecture in Liu and Xu's earlier work with Kemperman--Scherk Theorem. The proximity theorem combined with approaches in earlier work gives an FPT algorithm that finds a min-cost $F$-avoiding base parameterized by $|G|$. We additionally give an FPT algorithm parameterized by $|F|$, answering a question of Hörsch et al.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32798",
  "author_total": 3,
  "authors": [
    {
      "name": "Siyue Liu"
    },
    {
      "name": "Taihei Oki"
    },
    {
      "name": "Chao Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/92sw3w9g5p6sj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32798",
  "fulltext_url": null,
  "id": "92sw3w9g5p6sj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32798",
  "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.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7223661613",
  "pdf_url": "https://arxiv.org/pdf/2610.32798",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/92sw3w9g5p6sj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32798",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32798"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Proximity Theorem and FPT Algorithm for Minimum Group-labeled Matroid Bases",
  "title_text": "Proximity Theorem and FPT Algorithm for Minimum Group-labeled Matroid Bases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/92sw3w9g5p6sj",
  "venue": "arXiv",
  "versions": []
}