| import OAI.Computability.LoopMatching.Model |
|
|
| namespace OAI |
|
|
| /-! Polynomial-time approximation of weighted matchings with singleton loops. -/ |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace LoopMatching |
|
|
| def emptyPartialMatching (n : ℕ) : PartialMatching n := |
| ⟨∅, by |
| intro v e he |
| simp at he⟩ |
|
|
| @[simp] theorem loopVertices_emptyPartialMatching (n : ℕ) : |
| loopVertices (emptyPartialMatching n) = Finset.univ := by |
| classical |
| ext v |
| simp [loopVertices, emptyPartialMatching] |
|
|
| theorem partialMatching_eq_empty_of_le_one {n : ℕ} (hn : n ≤ 1) |
| (M : PartialMatching n) : M = emptyPartialMatching n := by |
| apply Subtype.ext |
| ext e |
| have hleft := e.val.1.isLt |
| have hright := e.val.2.isLt |
| have horder := e.property |
| change e.val.1.val < e.val.2.val at horder |
| omega |
|
|
| theorem count_eq_loop_product_of_le_one (G : Input) (hn : G.n ≤ 1) : |
| count G = ∏ v : Fin G.n, loopMultiplicity G v := by |
| classical |
| unfold count |
| rw [Finset.sum_eq_single (emptyPartialMatching G.n)] |
| · simp [weight, emptyPartialMatching, loopVertices] |
| · intro M _ hne |
| exact (hne (partialMatching_eq_empty_of_le_one hn M)).elim |
| · intro hnot |
| exact (hnot (Finset.mem_univ _)).elim |
|
|
| theorem count_order_zero (G : Input) (hn : G.n = 0) : count G = 1 := by |
| rw [count_eq_loop_product_of_le_one G (by omega)] |
| apply Finset.prod_eq_one |
| intro v _ |
| have hv := v.isLt |
| omega |
|
|
| theorem count_order_one (G : Input) (hn : G.n = 1) : |
| count G = loopMultiplicity G ⟨0, by omega⟩ := by |
| classical |
| rw [count_eq_loop_product_of_le_one G (by omega)] |
| apply Finset.prod_eq_single (⟨0, by omega⟩ : Fin G.n) |
| · intro v _ hne |
| have hv := v.isLt |
| have heq : v = (⟨0, by omega⟩ : Fin G.n) := by |
| apply Fin.ext |
| change v.val = 0 |
| omega |
| exact (hne heq).elim |
| · intro hnot |
| exact (hnot (Finset.mem_univ _)).elim |
|
|
| end LoopMatching |
|
|
| end |
|
|
| end OAI |
|
|
