{
  "abstract": "One item set evaluated at two capacities $R<D$ produces two concave hulls of optimal value against cardinality. We ask which prescribed pairs arise. Exchange arguments give a necessary system on vertex witnesses, exchange closure (EC), whose scalar consequences form the linear closure. We exhibit a pair satisfying every scalar test that fails EC, so the linear closure is strictly larger already at larger terminal count three; and a globally coherent EC witness admitting no common-size representation although its target pair is realizable. Under a cardinality cap, a four-band classification of one family gives exact thresholds for cap-four realizability, uncapped realizability and the vertex-only capped closure. Pairs whose larger terminal count is at most two are characterized. Under the explicit encoding the decision problem lies in $Σ_2^p$ and is polynomial-time for fixed terminal cardinalities. Exact finite certificates establish agreement of the scalar and witness conditions on six specified domains; sufficiency of uncapped EC remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28128",
  "author_total": 2,
  "authors": [
    {
      "name": "Prashant Chaudhary"
    },
    {
      "name": "Kapil Khandelwal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5cb7tz3q2t6q0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28128",
  "fulltext_url": null,
  "id": "5cb7tz3q2t6q0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28128",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.28128",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5cb7tz3q2t6q0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28128",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28128"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Inverse knapsack at two capacities: which pairs of value-cardinality hulls are realizable?",
  "title_text": "Inverse knapsack at two capacities: which pairs of value-cardinality hulls are realizable?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/5cb7tz3q2t6q0",
  "venue": "arXiv",
  "versions": []
}