{
  "abstract": "Selecting vertices covers their incident edges, but a matroid can prevent selecting the combinations that cover the most weight. We study whether pair-selection information improves rounding from a semidefinite relaxation of this problem. A feasible point $(x,Y)$ records individual and paired selection values. Its relaxation objective is $U$, and the matrix $Y-xx^\\top$ has rank $d$. We compare the feasible marginal vectors obtained by fixing one element as selected or excluded, then round only the best vector. For $d\\ge1$, the resulting basis has value at least $(3/4+1/(4d))U$. The procedure is deterministic and polynomial-time for a supplied exact rational point and an independence oracle. Weighting the covariance spectrum by the objective gives a stronger version of the bound. To show the rank factor is sharp, we construct a rank-$2m$ rational linear matroid and $m$ disjoint four-cycles with integral optimum $3m+1$, relaxation optimum $4m$, and an optimal covariance matrix of rank $m$. Thus the relaxation has limiting reciprocal integrality gap $4/3$, despite its conditional constraints. One valid pair-exclusion inequality makes it exact on this family, as an explicit semidefinite certificate shows. The result concerns rounding and the strength of this relaxation; it does not supply a low-rank optimal point or a fixed universal approximation improvement.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17512",
  "author_total": 1,
  "authors": [
    {
      "name": "Navid Rezazadeh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bph08ejcrqewp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17512",
  "fulltext_url": null,
  "id": "bph08ejcrqewp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17512",
  "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.OC"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7221753393",
  "pdf_url": "https://arxiv.org/pdf/2610.17512",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bph08ejcrqewp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17512",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17512"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Covariance-Rank Rounding and a Sharp Integrality Gap for a Matroid Edge-Coverage Moment Lift",
  "title_text": "Covariance-Rank Rounding and a Sharp Integrality Gap for a Matroid Edge-Coverage Moment Lift",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/bph08ejcrqewp",
  "venue": "arXiv",
  "versions": []
}