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    "headline": "The Information Bottleneck Is NP-Hard",
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    "dek": "Even approximating the optimum within a shrinking additive error remains hard unless P equals NP.",
    "summary": "Even approximating the optimum within a shrinking additive error remains hard unless P equals NP.",
    "body_text": "The information bottleneck seeks a compressed representation of one variable that preserves information about another. This paper proves that computing both the constrained optimum and its Lagrangian relaxation is NP-hard. Unless P equals NP, neither can be approximated in polynomial time within additive error on the order of the input alphabet size to the negative three-halves. The result concerns worst-case computation, not every practical heuristic.",
    "why_it_matters": "Even approximating the optimum within a shrinking additive error remains hard unless P equals NP.",
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      "The result concerns worst-case computation, not every practical heuristic."
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        "qualification": "The result concerns worst-case computation, not every practical heuristic."
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    "tags": [
      "information theory",
      "complexity",
      "representation learning"
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  "sources": [
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      "source_id": "source-2026-10-18-017",
      "title": "arXiv preprint 2610.22272, original v1",
      "publisher": "arXiv",
      "url": "https://arxiv.org/abs/2610.22272v1",
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    "title": "The Information Bottleneck Is NP-Hard",
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