File size: 1,651 Bytes
dae740b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | import Mathlib
namespace OAI
noncomputable section
namespace BoundedTreewidthL1
universe u_1
variable {V : Type u_1} (G : SimpleGraph V)
/-- Sum of the given positive real edge lengths along a graph walk. -/
def walkLength (ℓ : G.edgeSet → ℝ) : {u v : V} → G.Walk u v → ℝ
| _, _, .nil => 0
| u, _, .cons (v := v) h p => ℓ ⟨s(u, v), h⟩ + walkLength ℓ p
noncomputable def shortestPathDistance (ℓ : G.edgeSet → ℝ) (u v : V) : ℝ :=
sInf {r : ℝ | ∃ p : G.Walk u v, p.IsPath ∧ walkLength G ℓ p = r}
def HasTreeDecomposition [DecidableEq V] (k : ℕ) : Prop :=
∃ n : ℕ, ∃ T : SimpleGraph (Fin n), ∃ bag : Fin n → Finset V,
T.IsTree ∧
(∀ v : V, ∃ t, v ∈ bag t) ∧
(∀ u v : V, G.Adj u v → ∃ t, u ∈ bag t ∧ v ∈ bag t) ∧
(∀ v : V, (T.induce {t | v ∈ bag t}).Connected) ∧
(∀ t, (bag t).card ≤ k)
/-- A map to genuine finite-dimensional real ℓ₁, not just a cut pseudometric. -/
def HasL1Embedding (ℓ : G.edgeSet → ℝ) (C : ℝ) : Prop :=
∃ m : ℕ, ∃ F : V → PiLp (1 : ENNReal) (fun _ : Fin m => ℝ),
∀ u v : V,
shortestPathDistance G ℓ u v ≤ ‖F u - F v‖ ∧
‖F u - F v‖ ≤ C * shortestPathDistance G ℓ u v
def MainStatement : Prop :=
∀ k : ℕ, 2 ≤ k → ∃ C : ℝ, 1 ≤ C ∧
∀ (V : Type) [Fintype V] [DecidableEq V] [Nonempty V]
(G : SimpleGraph V),
G.Connected → HasTreeDecomposition G k →
∀ ℓ : G.edgeSet → ℝ, (∀ e, 0 < ℓ e) → HasL1Embedding G ℓ C
theorem main_theorem : MainStatement := by
sorry
end BoundedTreewidthL1
end
end OAI
|