{
  "abstract": "Given a joint distribution $P_{XY}$ of random variables $X,Y$ on finite alphabets, the information bottleneck method seeks a representation $T$ of $X$ that is as informative as possible about $Y$ while compressing $X$; the privacy funnel asks for a representation that discloses much about $X$ while leaking little about $Y$. In both scenarios $T$ and $Y$ are conditionally independent given $X$. We show that computing the optimal value $\\operatorname{IB}(P_{XY},R) = \\sup_T \\{ I(Y;T) : I(X;T) \\le R \\}$ of the information bottleneck problem, as well as its Lagrangian relaxation $L_β(P_{XY}) = \\sup_T [ I(Y;T) - βI(X;T) ]$, is NP-hard. Moreover, unless $\\mathrm{P} = \\mathrm{NP}$, neither $\\operatorname{IB}(P_{XY},R)$ nor $L_β(P_{XY})$ can be approximated in polynomial time within an additive error of order $|\\mathcal{X}|^{-(1+α)}$, for every fixed $α> 0$. We also prove that computing the privacy funnel $\\operatorname{PF}(P_{XY},r) = \\inf_T \\{ I(Y;T) : I(X;T) \\ge r \\}$ and its Lagrangian relaxation $Λ_β(P_{XY}) = \\inf_T [ I(Y;T) - βI(X;T) ]$ is NP-hard, and that, unless $\\mathrm{P} = \\mathrm{NP}$, neither can be approximated in polynomial time within an additive error of order $|\\mathcal{X}|^{-2}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22272",
  "author_total": 2,
  "authors": [
    {
      "name": "Roberto Bruno"
    },
    {
      "name": "Ugo Vaccaro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dbsmmjd1ha5xq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22272",
  "fulltext_url": null,
  "id": "dbsmmjd1ha5xq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22272",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7222663746",
  "pdf_url": "https://arxiv.org/pdf/2610.22272",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dbsmmjd1ha5xq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22272",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22272"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "The Information Bottleneck is NP-hard",
  "title_text": "The Information Bottleneck is NP-hard",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/dbsmmjd1ha5xq",
  "venue": "arXiv",
  "versions": []
}