{
  "external_premise": {
    "statement": "ACYCLIC TRANSITIVITY EDITING is NP-hard.",
    "source": "Weller, Komusiewicz, Niedermeier, Uhlmann (2012), cited by arXiv:2606.00278 Section 3.2",
    "reproduced_here": false,
    "why": "A hardness lower bound quantifies over all algorithms; it cannot be established by running one. It is inherited from the cited proof."
  },
  "rows": [
    {
      "n": 3,
      "domain": "exhaustive: all acyclic digraphs",
      "n_graphs": 25,
      "n_transitively_closed_dags": 19,
      "mismatches": 0,
      "control_plain_differs": 0,
      "max_editing_optimum": 1
    },
    {
      "n": 4,
      "domain": "exhaustive: all acyclic digraphs",
      "n_graphs": 543,
      "n_transitively_closed_dags": 219,
      "mismatches": 0,
      "control_plain_differs": 0,
      "max_editing_optimum": 1
    },
    {
      "n": 5,
      "domain": "random sample of 300 acyclic digraphs",
      "n_graphs": 300,
      "n_transitively_closed_dags": 4231,
      "mismatches": 0,
      "control_plain_differs": 0,
      "max_editing_optimum": 2
    }
  ],
  "reduction_polynomial_time": "G -> G+ adds C(n,2) bidirected edges: O(n^2) time, so the reduction is polynomial.",
  "conclusion": "Given the external premise and the exhaustively verified reduction, computing incomp(G) is NP-hard."
}