{
  "schema_version": 1,
  "problem_number": "OWR-2489-009",
  "title": "Counterexamples to Conforti's Subtree Conjecture for Mixed-Integer Bipartite Covers",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a bipartite graph G = (U ∪ V, E), a set I ⊆ U ∪ V and rationals b_ij, let S(G,I) = {x ∈ R^(U∪V) : x_i + x_j ≥ b_ij (ij ∈ E), x_i ∈ Z (i ∈ I)}, and let k be the least positive integer with kb integral. In the 2008 Oberwolfach report on combinatorial optimization, Conforti conjectured that conv S(G,I) is the intersection of the hulls conv S(T, I ∩ V(T)) over the subtrees T of G whose integral vertices are exactly their leaves. This would place the membership problem for conv S(G,I) in coNP. He noted that the case k = 2 follows from work of Conforti, Gerards and Zambelli, and that the conjecture was open for every k ≥ 3. We show that it fails for every k ≥ 3. For each such k we give two unicyclic counterexamples. One has six vertices. The other has seven, and in it every integral vertex is a pendant vertex with a continuous neighbour, so the standard normalisation of such sets (splitting integral vertices) does not remove it. In both, an explicit point lies in conv S(T, I ∩ V(T)) for every subtree T of G, but violates a facet-defining inequality of conv S(G,I) by (k − 2)/(2k − 3). The proofs are short and by hand. We also describe exact validity certificates based on an extended formulation, and use them to certify a further normalised counterexample for k = 3. The complexity of the membership problem remains open. This is an unrefereed note.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.OC",
    "math.CO",
    "cs.DM"
  ],
  "keywords": [
    "mixed-integer programming",
    "convex hull",
    "bipartite graphs",
    "mixed-integer covers",
    "network-dual sets",
    "extended formulation",
    "counterexample",
    "Conforti's conjecture",
    "Oberwolfach Reports",
    "OWR-2489-004",
    "OWR-2489-009",
    "math.OC",
    "math.CO",
    "cs.DM",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-2489-009/",
  "pdf_url": "https://eulersolve.org/papers/owr-2489-009/paper.pdf?v=8f2c5fd65e65",
  "doi": "10.5281/zenodo.23062398",
  "zenodo_record_url": "https://zenodo.org/records/23062398",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Refutes Conforti's Conjecture 2 (OWR 51/2008; corpus records OWR-2489-004 and OWR-2489-009) for every k ≥ 3, also for normalised instances. The case k = 2 is due to Conforti, Gerards and Zambelli and is not re-derived here; the complexity of the membership problem (record OWR-2489-008) remains open.",
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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."
}