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8df23f4 | 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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | import OAI.Computability.BinPacking.Inventory.IntegralSpeciesCounts
import OAI.Computability.BinPacking.Inventory.SpeciesAllocation
import OAI.Computability.BinPacking.Packing.ConstructLocalState
import OAI.Computability.BinPacking.Packing.ExtractedCover
import OAI.Computability.BinPacking.Packing.LocalTemplateBases
import OAI.Computability.BinPacking.Packing.PhysicalRolePacking
namespace OAI
noncomputable section
namespace BinPackingGap.InventoryData
open scoped BigOperators
variable (D : InventoryData)
def integralGlobalAllocation (C : Finset D.Vertex) (hC : C.card = D.k) :
D.PhysicalBins ≃ D.GlobalCopy := by
classical
exact SpeciesAllocation.allocation (D.integralGlobalSpecies C) D.globalStock
(fun s => by rw [Fintype.card_subtype]; exact D.card_integralGlobalSpecies C hC s)
def integralFlagAllocation : D.PhysicalBins ≃ D.FlagCopy := by
classical
exact SpeciesAllocation.allocation D.integralFlagSpecies D.flagStock
(fun s => by rw [Fintype.card_subtype]; exact D.card_integralFlagSpecies s)
@[simp] theorem integralGlobalAllocation_species (C : Finset D.Vertex)
(hC : C.card = D.k) (b : D.PhysicalBins) :
(D.integralGlobalAllocation C hC b).1 = D.integralGlobalSpecies C b :=
SpeciesAllocation.allocation_species _ _ _ b
@[simp] theorem integralFlagAllocation_species (b : D.PhysicalBins) :
(D.integralFlagAllocation b).1 = D.integralFlagSpecies b :=
SpeciesAllocation.allocation_species _ _ _ b
def integralRoleEquiv (C : Finset D.Vertex) (hC : C.card = D.k)
(S : ∀ v, D.LocalState v (D.coverSelected C v)) :
∀ r : Role, D.PhysicalBins ≃ D.RoleCopies r
| .x => Equiv.sigmaCongrRight fun v => D.stateRowEquiv v (D.coverSelected C v)
| .anchor => Equiv.sigmaCongrRight fun v => (S v).anchorAtEquiv
| .«global» => D.integralGlobalAllocation C hC
| .«local» => Equiv.sigmaCongrRight fun v => (S v).localAtEquiv
| .flag => D.integralFlagAllocation
theorem integralRoleEquiv_capacity (C : Finset D.Vertex)
(hC : C.card = D.k) (hcover : D.graph.IsCover C)
(S : ∀ v, D.LocalState v (D.coverSelected C v)) (bin : D.PhysicalBins) :
(∑ r : Role, D.itemSize ⟨r, D.integralRoleEquiv C hC S r bin⟩) ≤ 1 := by
rcases bin with ⟨v, b | j⟩
· change (∑ r, D.itemSize (D.localTemplateItems
(D.stateMainRow v (D.coverSelected C v) b)
(.inl ((S v).anchorEquiv b))
(D.integralGlobalAllocation C hC ⟨v, .inl b⟩)
(.inl ((S v).localEquiv b))
(D.integralFlagAllocation ⟨v, .inl b⟩) r)) ≤ 1
exact (D.stateMainTemplate_feasible_iff (D.coverSelected C v) b
((S v).anchorEquiv b) ((S v).localEquiv b) _ _
(D.integralGlobalAllocation_species C hC ⟨v, .inl b⟩)
(D.integralFlagAllocation_species ⟨v, .inl b⟩)).mpr
((S v).completion_le_deadline b)
· change (∑ r, D.itemSize (D.localTemplateItems
(.inr (D.coverSelected C v, j)) (.inr j)
(D.integralGlobalAllocation C hC ⟨v, .inr j⟩) (.inr j)
(D.integralFlagAllocation ⟨v, .inr j⟩) r)) ≤ 1
apply (D.edgeTemplate_feasible_iff (D.coverSelected C v)
(D.integralPermit C j.1.1.val.1 j.1.1.val.2) j _ _
(D.integralGlobalAllocation_species C hC ⟨v, .inr j⟩)
(D.integralFlagAllocation_species ⟨v, .inr j⟩)).mpr
cases hs : D.coverSelected C v
· exact Or.inr (D.integral_edge_permit_of_unselected C hcover j hs)
· exact Or.inl rfl
theorem hasPacking_of_localStates (C : Finset D.Vertex) (hC : C.card = D.k)
(hcover : D.graph.IsCover C)
(S : ∀ v, D.LocalState v (D.coverSelected C v)) :
HasPacking D.packingInstance D.B := by
have p := D.packingOfRoleEquivs (D.integralRoleEquiv C hC S)
(D.integralRoleEquiv_capacity C hC hcover S)
simpa only [HasPacking, D.card_physicalBins] using (Nonempty.intro p)
theorem ofCompetingTrees_hasPacking (G : GraphInput) {d : ℕ}
(T : CompetingTrees d) (hd : 0 < d) (R k : ℕ)
(hk : k ≤ G.n) (hcover : G.CoverAtMost k) :
HasPacking (ofCompetingTrees G T R k).packingInstance
(ofCompetingTrees G T R k).B := by
obtain ⟨C, hcover, hC⟩ := hcover.exists_exact hk
exact (ofCompetingTrees G T R k).hasPacking_of_localStates C hC hcover
(fun v => constructedLocalState G T hd R k v _)
end BinPackingGap.InventoryData
end
namespace BinPackingGap
theorem cover_hasPacking_reduction (c : ℕ) (rho : ℝ) (G : GraphInput) (k : ℕ)
(hk : k ≤ G.n) (hcover : G.CoverAtMost k) :
HasPacking (reductionInstance c rho G k) (reductionBinBound c rho G k) :=
InventoryData.ofCompetingTrees_hasPacking G (reductionTrees c rho)
(depthDemand_pos c rho) _ k hk hcover
end BinPackingGap
end OAI
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