{
  "schema_version": 1,
  "problem_number": "OWR-730-009",
  "title": "The Exact Complexity of ε-Dense Steiner Tree",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In the ε-Dense Steiner Tree problem of Karpinski and Zelikovsky, every terminal is adjacent to at least an ε-fraction of the non-terminals, and a Steiner tree with the fewest edges is sought. For every fixed ε > 0 the problem has a polynomial-time approximation scheme. At an Oberwolfach problem session in 2004, Hauptmann asked for hardness results and noted that it was not even known whether the exact problem is NP-hard; the question was still described as open in 2015 and in 2020. We show that for every fixed ε ∈ (0,1] the problem can be solved exactly in time n^{O(log n/ε)}. The main step is a structural lemma: if H is any set of non-terminals that are all adjacent to terminals, and G[S ∪ H] has r components, then every optimal tree has at most |H| + 2r − 2 Steiner vertices adjacent to terminals. Consequently the problem is not NP-hard, even under Turing reductions, unless NP ⊆ DTIME(2^{O(log² n)}). Conversely, for every fixed ε ∈ (0,1), a reduction from 3-SAT in the style of Megiddo and Vishkin, combined with a dense covering gadget over F_q^d, shows that the problem has no N^{o(log N)}-time algorithm unless the Exponential Time Hypothesis (ETH) fails, and that it is not in P unless FPT = W[2]. Hence, assuming ETH, exact ε-Dense Steiner Tree is neither in P nor NP-hard. Of the two halves of this statement, \"not NP-hard\" needs only NP ⊄ QP, while \"not in P\" needs ETH or FPT ≠ W[2]. This is an unrefereed note.",
  "result_type": "COMPLETE_CONDITIONAL_RESOLUTION",
  "categories": [
    "cs.CC",
    "cs.DS",
    "math.CO"
  ],
  "keywords": [
    "Steiner tree",
    "dense instances",
    "exact algorithms",
    "quasi-polynomial time",
    "Exponential Time Hypothesis",
    "W[2]-hardness",
    "NP-hardness",
    "parameterized complexity",
    "Oberwolfach Reports",
    "OWR-730-009",
    "cs.CC",
    "cs.DS",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-29",
  "publication_date": "2026-09-29",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-29",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-730-009/",
  "pdf_url": "https://eulersolve.org/papers/owr-730-009/paper.pdf?v=e51b928e2310",
  "doi": "10.5281/zenodo.23041942",
  "zenodo_record_url": "https://zenodo.org/records/23041942",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Resolves Hauptmann's question in the only form possible without separating P from NP: the exact problem is not NP-hard unless NP ⊆ QP, and not in P unless the Exponential Time Hypothesis fails (or FPT = W[2]). It is not an unconditional resolution. Unrefereed.",
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    "source.zip": {
      "sha256": "7122e5d8162fe4d3c7f1934b9d4773942bab2456019f381ee352bd7fc80041d6"
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}