{
  "abstract": "We study the problem of maximizing a linear function over the integer points of an origin-centered $\\ell_p$ ball, which we call \\BallIPp{p}. For every fixed integer $p\\ge2$, we prove that the decision problem over an $\\ell_p$-ball is NP-complete. We then focus on the Euclidean case and study how the difficulty of the problem depends on the numerical parameters of the instance. We give two complementary pseudo-polynomial exact algorithms. The first specializes a known radius-budget dynamic program; the same nonlinear-knapsack framework also applies to every fixed finite integer $p$. We then develop a complementary dynamic program over candidate objective values for the Euclidean case. The latter is polynomial in the encoding size of the radius and pseudo-polynomial in the magnitude of the cost coefficients. For a fixed objective value, feasibility can be formulated as a closest vector problem (CVP) instance. This connection gives an exact algorithm whose running time depends on the covering radius of that lattice. Conversely, we show that a rank-$k$ Euclidean closest-vector instance in ambient dimension $d$ reduces to the decision version of a Euclidean \\BallIP{} instance in dimension $d+k\\le2d$, transferring known bounds under the Exponential Time Hypothesis (ETH). Finally, we study dimension-dependent approaches based on proximity to the continuous optimizer and describe the fixed-level constructions that extend to ellipsoids.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15114",
  "author_total": 2,
  "authors": [
    {
      "name": "Cinar Ari"
    },
    {
      "name": "Robert Hildebrand"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9mvqvke4avy5w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15114",
  "fulltext_url": null,
  "id": "9mvqvke4avy5w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15114",
  "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": "W7213335696",
  "pdf_url": "https://arxiv.org/pdf/2609.15114",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9mvqvke4avy5w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15114",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15114"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Integer Maximization over $\\ell_p$ Balls: Hardness and Exact Algorithms",
  "title_text": "Integer Maximization over ℓ_p Balls: Hardness and Exact Algorithms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/9mvqvke4avy5w",
  "venue": "arXiv",
  "versions": []
}