{
  "schema_version": 1,
  "problem_number": "AIM-COMPUTATION-0021",
  "title": "Minimum Support Augmentation in Chordal Graphical Models",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Given an observed support in a finite Cartesian table, we minimize the number of distinct new cells needed to make every maximal-clique projection of a chordal graphical model surjective. For a clique path, the missing clique configurations form a partially ordered set, and the minimum augmentation equals its width. This gives an exact algorithm by bipartite matching, as well as an equivalent lower-bound circulation formulation. In contrast, the decision problem is NP-complete on the fixed six-variable chordal graph with maximal cliques ABC, ABX, ACY, and BCZ, even when X, Y, Z are binary; the alphabets of A, B, C are allowed to grow. A partial Latin square supplies the reduction and a concrete support for which the three individual root-leaf optima are five but the global optimum is six. Integer separator multiplicities also give an exact dynamic program on any clique tree. The statistical interpretation uses the classical positive-MLE criterion for decomposable models. The explicitly stated augmentation results are complete; the general AIM marginal-cone facet question is not resolved. This AI-assisted, self-audited preprint is unrefereed and makes no claim of absolute priority.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.ST",
    "math.CO",
    "cs.DS"
  ],
  "keywords": [
    "chordal graphical model",
    "support augmentation",
    "maximum likelihood",
    "clique path",
    "Dilworth theorem",
    "bipartite matching",
    "NP-completeness",
    "partial Latin square",
    "AIM-COMPUTATION-0021"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-computation-0021/",
  "pdf_url": "https://eulersolve.org/papers/aim-computation-0021/paper.pdf?v=f1f97b9f50e9",
  "doi": "10.5281/zenodo.23380361",
  "zenodo_record_url": "https://zenodo.org/records/23380361",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Full finite Cartesian cell space with no structural zeros and nonempty observed support. Complete proofs of the stated augmentation theorems, not a resolution of the general AIM marginal-cone facet question or a graph dichotomy. NP-completeness permits growing alphabets; the general DP is polynomial for fixed clique configuration counts, not treewidth alone. Classical MLE, Dilworth, bag gluing, Latin completion hardness and prior cover results are credited. Self-audited, unrefereed, AI-assisted; no independent review, formal verification or absolute priority claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "f1f97b9f50e90e19ee9b83b220332eee265563e5e1cfcd26c40fc041fd551d2d"
    },
    "source.zip": {
      "sha256": "3c7d8a69654e1a0809849986f98be016deb3ccb7b7bebf785242047615f86e04"
    },
    "verification_report.md": {
      "sha256": "728e7f470eb07981a754a1d1e19855195c434cb394a250fef2c211e6f68e3a7e"
    }
  },
  "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."
}