{
  "abstract": "We study two inclusion-wise extremal variants of the 0-1 Knapsack Problem (KP). Given $n$ items with integer weights and nonnegative costs or profits, the Minimum-Cost Maximal Knapsack Packing Problem seeks a minimum-cost packing for capacity $c$ to which no item can be added, while the Maximum-Profit Minimal Knapsack Cover Problem seeks a maximum-profit cover for threshold $d$ from which no item can be removed. Both problems are weakly $\\mathrm{NP}$-hard and pseudo-polynomially solvable, as is the classical KP, but their approximability is fundamentally different: unless $\\mathrm{P}=\\mathrm{NP}$, with positive values neither problem admits a polynomial-time $2^{p(n)}$-approximation for any polynomial $p$; allowing zero values rules out every multiplicative factor. Even with unit costs for packing and unit profits for covering, neither admits an $n^{1-\\varepsilon}$-approximation for any fixed $\\varepsilon>0$, whereas a trivial $n$-approximation exists. We identify restrictions that restore approximability. If every item weighs at least a fixed fraction of the capacity, respectively of the threshold, both problems can be solved exactly in polynomial time. For the cover problem, if each weight $w_i$ satisfies $w_i-1\\geδ(d-1)$, we give an approximation scheme for arbitrary profits whose running time is polynomial in $n$, $1/\\varepsilon$, and $1/δ$. If values equal weights, a fully polynomial-time approximation scheme (FPTAS) was already known for the packing problem; we give an FPTAS for the cover problem, improving on the previously known factor $2$. If value-to-weight ratios differ by at most a factor $κ$, we obtain a $(1+\\varepsilon)κ$-approximation for both problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18017",
  "author_total": 4,
  "authors": [
    {
      "name": "Fabio Ciccarelli"
    },
    {
      "name": "Fabio Furini"
    },
    {
      "name": "Ulrich Pferschy"
    },
    {
      "name": "Elena Rener"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5rtc6kcjee9k1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18017",
  "fulltext_url": null,
  "id": "5rtc6kcjee9k1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18017",
  "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.OC"
    ],
    [
      "openalex",
      "2209|Optimization and Packing Problems"
    ]
  ],
  "openalex": "W7221750661",
  "pdf_url": "https://arxiv.org/pdf/2610.18017",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5rtc6kcjee9k1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.18017",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18017"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Inclusion-wise extremal knapsack problems are (mostly) hard to approximate",
  "title_text": "Inclusion-wise extremal knapsack problems are (mostly) hard to approximate",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/5rtc6kcjee9k1",
  "venue": "arXiv",
  "versions": []
}