File size: 3,792 Bytes
97c3405 | 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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.Combinatorics.ContingencyTables.Machines.CellMaximumProgram
import OAI.Combinatorics.ContingencyTables.Machines.BoundedComplementProgram
namespace OAI
/-! # Computing an attained minimum through the complementary table
Complementation changes the lower endpoint query to the
upper endpoint query and preserves every cell capacity.
-/
namespace ContingencyTables.CellMinimumProgram
open MatchingFPRAS.TreeTyped CellLowerTestProgram
def complemented (z : Context) : Context := (BoundedComplementProgram.complement z.1,z.2)
def complementedRealizer : Realizer complemented :=
pair (composition (f := fun z : Context=>z.1)
(g := BoundedComplementProgram.complement) first BoundedComplementProgram.complementRealizer) second
lemma polynomial_complemented : PolynomialTime complementedRealizer :=
polynomial_pair (polynomial_composition polynomial_first
BoundedComplementProgram.polynomial_complement) polynomial_second
def minimum (z : Context) : ℕ :=
CellMaximumProgram.bound z-CellMaximumProgram.maximum (complemented z)
def minimumRealizer : Realizer minimum :=
composition (f := fun z : Context=>
(CellMaximumProgram.bound z,CellMaximumProgram.maximum (complemented z)))
(g := fun z : ℕ×ℕ=>z.1-z.2)
(pair CellMaximumProgram.boundRealizer
(composition (f := complemented) (g := CellMaximumProgram.maximum)
complementedRealizer CellMaximumProgram.maximumRealizer)) subtract
lemma polynomial_minimum : PolynomialTime minimumRealizer :=
polynomial_composition (polynomial_pair CellMaximumProgram.polynomial_bound
(polynomial_composition polynomial_complemented CellMaximumProgram.polynomial_maximum))
polynomial_subtract
lemma complemented_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ)
(i : Fin m) (j : Fin n) :
complemented (NetworkVertexCode.data r c b,(i.val,j.val))=
(NetworkVertexCode.data (BoundedTableComplement.rows b r)
(BoundedTableComplement.columns b c) b,(i.val,j.val)) := by
unfold complemented
rw [BoundedComplementProgram.complement_code]
/-- The returned lower endpoint is attained and is no larger than the
selected coordinate of any feasible bounded table. -/
theorem minimum_correct {m n : ℕ} (hEK : PublishedEdmondsKarp.EdmondsKarp1972Theorem1)
(r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ)
(i : Fin m) (j : Fin n)
(hne : ∃ X,HasMargins X r c ∧ ∀ u v,X u v≤b u v) :
∃ X,HasMargins X r c ∧ (∀ u v,X u v≤b u v) ∧
X i j=minimum (NetworkVertexCode.data r c b,(i.val,j.val)) ∧
∀ Y,HasMargins Y r c → (∀ u v,Y u v≤b u v) → X i j≤Y i j := by
obtain ⟨X₀,hx,hxb⟩ := hne
have hn : ∃ Y,HasMargins Y (BoundedTableComplement.rows b r)
(BoundedTableComplement.columns b c) ∧ ∀ u v,Y u v≤b u v :=
⟨BoundedTableComplement.complement b X₀,
BoundedTableComplement.complement_margins b X₀ r c hx hxb,
BoundedTableComplement.complement_bounds b X₀⟩
obtain ⟨Y,hy,hyb,hvalue,hmax⟩ := CellMaximumProgram.maximum_correct hEK
(BoundedTableComplement.rows b r) (BoundedTableComplement.columns b c) b i j hn
refine ⟨BoundedTableComplement.complement b Y,
BoundedTableComplement.complement_reverse_margins b X₀ Y r c hx hxb hy hyb,
BoundedTableComplement.complement_bounds b Y,?_,?_⟩
· unfold minimum
rw [CellMaximumProgram.bound_code,complemented_code,←hvalue]
rfl
· intro Z hz hzb
have hb := hmax (BoundedTableComplement.complement b Z)
(BoundedTableComplement.complement_margins b Z r c hz hzb)
(BoundedTableComplement.complement_bounds b Z)
change b i j-Y i j≤Z i j
change b i j-Z i j≤Y i j at hb
omega
end ContingencyTables.CellMinimumProgram
end OAI
|