| import OAI.Computability.LoopMatching.MachineMain |
| import OAI.Computability.LoopMatching.ExpandCount |
| import OAI.Computability.LoopMatching.SparseBoundary |
|
|
| namespace OAI |
|
|
| /-! Polynomial-time approximation of weighted matchings with singleton loops. -/ |
|
|
| namespace LoopMatching |
|
|
| theorem actual_algorithm_correct (G : Input) : |
| 0 < (MachineMain.A G).2 ∧ |
| (MachineMain.A G).1 ≤ (MachineMain.A G).2 * count G ∧ |
| (MachineMain.A G).2 * count G ≤ 2 ^ (18 * G.n) * (MachineMain.A G).1 ∧ |
| ((MachineMain.A G).1 = 0 ↔ count G = 0) := by |
| by_cases h : 2 * G.ordinary.records.length + G.loops.length < G.n |
| · have hz := count_eq_zero_of_sparse_bound G h |
| simp [MachineMain.A, h, hz] |
| · rw [MachineMain.A, ite_eq_right h] |
| exact ⟨Approximation.output_den_pos G, |
| Approximation.output_bounds_of_expanded_count G |
| (fun j _ => expanded_count G j)⟩ |
|
|
| theorem fullEndpoint : FullEndpoint := by |
| obtain ⟨machine, hfinite, hlength⟩ := MachineMain.computational_endpoint |
| exact ⟨MachineMain.A, machine, hfinite, hlength, actual_algorithm_correct⟩ |
|
|
| end LoopMatching |
|
|
| end OAI |
|
|
