{
  "abstract": "Weakly submodular maximization finds numerous applications in machine learning, data mining and statistics. In particular, under a cardinality constraint, the standard greedy algorithm is known to attain the tight $(1-e^{-γ})$-approximation guarantee, where $γ$ denotes the submodularity ratio. However, when considering general matroid constraints, \\emph{how well standard greedy can perform} has remained a longstanding open question. In this paper, we resolve this question by studying the worst-case approximation guarantee $R_{\\mathrm{gr}}(γ,r)$ of standard forward greedy, where $r$ is the rank of matroid. Specifically, our first contribution is the following refined lower bound \\begin{equation*} R_{gr}(γ,r)\\ge \\max\\left\\{ \\fracγ{r}, \\frac{γ^2} {B_γ+\\sqrt{γ(r-1)}}, \\frac{γ^{r+1}} {2-γ+γ^{r+1}}, \\frac{γ^2} {14\\left(B_γ+\\sqrt{γ(1-γ)(r-1)}\\right)} \\right\\}, \\end{equation*} where $B_γ\\triangleq2-γ+γ^2$. This lower bound not only strictly improves upon the previous guarantee of $0.4γ^2/(1+\\sqrt{γr})$, but also can recover the classical $1/2$-approximation guarantee at $γ=1$. Subsequently, a corresponding upper bound is established for $R_{\\mathrm{gr}}(γ,r)$, namely, \\begin{equation}\\label{eq_upper_abstract} R_{gr}(γ,r) \\le\\min\\left\\{\\frac{1}{2},20\\max\\left\\{\\fracγ{r}, \\frac{γ^2}{B_γ+ \\sqrt{γ(1-γ)(r-1)}}\\right\\}\\right\\}. \\end{equation} For every fixed $γ\\in(0,1)$, this equation effectively sharpens the recent $O_γ(1/\\log(n-1))$ upper bound to $O_γ(r^{-1/2})$, where $n$ is the size of ground set. Putting all these results together, we eventually can obtain \\[R_{gr}(γ,r) = Θ\\left( \\max\\left\\{\\fracγ{r}, \\frac{γ^2} {1+\\sqrt{γ(1-γ)r}}\\right\\} \\right).\\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26967",
  "author_total": 5,
  "authors": [
    {
      "name": "Qixin Zhang"
    },
    {
      "name": "Shi Fu"
    },
    {
      "name": "Shengminjie Chen"
    },
    {
      "name": "Chao Qian"
    },
    {
      "name": "Dacheng Tao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6y7v717x4gv00.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26967",
  "fulltext_url": null,
  "id": "6y7v717x4gv00",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26967",
  "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.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7223083658",
  "pdf_url": "https://arxiv.org/pdf/2610.26967",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6y7v717x4gv00.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.26967",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26967"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "When Weak Submodularity Meets Matroids: How Does Standard Greedy Behave?",
  "title_text": "When Weak Submodularity Meets Matroids: How Does Standard Greedy Behave?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/6y7v717x4gv00",
  "venue": "arXiv",
  "versions": []
}