| import OAI.Combinatorics.PerfectMatching.Basic |
|
|
| namespace OAI |
|
|
| /-! |
| Degree and odd-cut constraints for the perfect matching polytope. |
| -/ |
|
|
| noncomputable section |
|
|
| open scoped BigOperators Topology |
| open Filter |
|
|
| namespace MatchingEntropy |
|
|
| variable {V E : Type*} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] |
|
|
| namespace LooplessGraph |
|
|
| / |
| def Crosses (G : LooplessGraph V E) (S : Finset V) (e : E) : Prop := |
| (G.left e ∈ S ∧ G.right e ∉ S) ∨ (G.left e ∉ S ∧ G.right e ∈ S) |
| |
| instance crossesDecidable (G : LooplessGraph V E) (S : Finset V) (e : E) : |
| Decidable (G.Crosses S e) := inferInstanceAs (Decidable |
| ((G.left e ∈ S ∧ G.right e ∉ S) ∨ (G.left e ∉ S ∧ G.right e ∈ S))) |
| |
| def cut (G : LooplessGraph V E) (S : Finset V) : Finset E := |
| Finset.univ.filter (G.Crosses S) |
| |
| def degree (G : LooplessGraph V E) (x : E → ℝ) (v : V) : ℝ := |
| ∑ e, if G.Incident v e then x e else 0 |
| |
| def cutMass (G : LooplessGraph V E) (x : E → ℝ) (S : Finset V) : ℝ := |
| ∑ e ∈ G.cut S, x e |
| |
| / |
| def constraintPolytope (G : LooplessGraph V E) : Set (E → ℝ) := |
| {x | (∀ e, 0 ≤ x e) ∧ (∀ v, G.degree x v = 1) ∧ |
| (∀ S : Finset V, Odd S.card → 1 ≤ G.cutMass x S)} |
| |
| omit [Fintype V] [DecidableEq E] in |
| @[simp] theorem mem_cut (G : LooplessGraph V E) (S : Finset V) (e : E) : |
| e ∈ G.cut S ↔ G.Crosses S e := by simp [cut] |
| |
| omit [Fintype E] [DecidableEq E] in |
| theorem crosses_compl (G : LooplessGraph V E) (S : Finset V) (e : E) : |
| G.Crosses Sᶜ e ↔ G.Crosses S e := by |
| simp only [Crosses, Finset.mem_compl, not_not] |
| tauto |
| |
| omit [DecidableEq E] in |
| @[simp] theorem cut_compl (G : LooplessGraph V E) (S : Finset V) : |
| G.cut Sᶜ = G.cut S := by ext; simp [crosses_compl] |
| |
| omit [Fintype V] in |
| theorem degree_indicator (G : LooplessGraph V E) (M : G.Matching) (v : V) : |
| G.degree (G.indicator M) v = 1 := by |
| obtain ⟨e, ⟨he, hev⟩, huniq⟩ := M.property v |
| rw [degree, Finset.sum_eq_single e] |
| · simp [hev, indicator, he] |
| · intro e' _ hne |
| by_cases hv : G.Incident v e' |
| · have hh : e' ∉ M.val := by |
| intro hm |
| exact hne (huniq e' ⟨hm, hv⟩) |
| simp [hv, indicator, hh] |
| · simp [hv] |
| · simp |
| |
| omit [Fintype V] in |
| theorem degree_mean (G : LooplessGraph V E) (p : G.Matching → ℝ) (v : V) : |
| G.degree (G.mean p) v = ∑ M, p M := by |
| classical |
| have he : ∀ e, (if G.Incident v e then G.mean p e else 0) = |
| ∑ M, p M*(if G.Incident v e then G.indicator M e else 0) := by |
| intro e |
| by_cases h : G.Incident v e <;> simp [h, mean] |
| unfold degree |
| simp_rw [he] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro M _ |
| rw [← Finset.mul_sum] |
| change p M*G.degree (G.indicator M) v = p M |
| rw [G.degree_indicator, mul_one] |
| |
| omit [Fintype V] [Fintype E] [DecidableEq E] in |
| theorem incidence_sum (G : LooplessGraph V E) (e : E) (S : Finset V) : |
| (∑ v ∈ S, if G.Incident v e then (1:ℕ) else 0) = |
| (if G.left e ∈ S then 1 else 0) + (if G.right e ∈ S then 1 else 0) := by |
| have hv : ∀ v, (if G.Incident v e then (1:ℕ) else 0) = |
| (if v = G.left e then 1 else 0) + (if v = G.right e then 1 else 0) := by |
| intro v |
| by_cases hl : v = G.left e <;> by_cases hr : v = G.right e |
| · exact ((G.loopless e) (hl.symm.trans hr)).elim |
| · simp [Incident, hl, G.loopless e, Ne.symm (G.loopless e)] |
| · simp [Incident, hr, G.loopless e, Ne.symm (G.loopless e)] |
| · simp [Incident, Ne.symm hl, Ne.symm hr, hl, hr] |
| simp_rw [hv, Finset.sum_add_distrib] |
| simp |
| |
| omit [Fintype V] [Fintype E] [DecidableEq E] in |
| theorem matching_incidence_count (G : LooplessGraph V E) (M : G.Matching) |
| (S : Finset V) : |
| (∑ e ∈ M.val, ((if G.left e ∈ S then (1:ℕ) else 0) + |
| (if G.right e ∈ S then 1 else 0))) = S.card := by |
| have hsum : ∀ v, (∑ e ∈ M.val, if G.Incident v e then (1:ℕ) else 0) = 1 := by |
| intro v |
| obtain ⟨e, ⟨he, hev⟩, hu⟩ := M.property v |
| rw [Finset.sum_eq_single e] |
| · simp [hev] |
| · intro e' he' hh |
| have hv : ¬G.Incident v e' := fun hv => hh (hu e' ⟨he', hv⟩) |
| simp [hv] |
| · simp [he] |
| calc |
| _ = ∑ e ∈ M.val, ∑ v ∈ S, if G.Incident v e then (1:ℕ) else 0 := by |
| simp_rw [incidence_sum] |
| _ = ∑ v ∈ S, ∑ e ∈ M.val, if G.Incident v e then (1:ℕ) else 0 := Finset.sum_comm |
| _ = S.card := by simp_rw [hsum]; simp |
| |
| omit [Fintype V] [Fintype E] [DecidableEq E] in |
| theorem matching_crosses_odd (G : LooplessGraph V E) (M : G.Matching) |
| (S : Finset V) (hS : Odd S.card) : ∃ e ∈ M.val, G.Crosses S e := by |
| by_contra h |
| push Not at h |
| have heq : ∀ e ∈ M.val, ((if G.left e ∈ S then (1:ℕ) else 0) + |
| (if G.right e ∈ S then 1 else 0)) = 2*(if G.left e ∈ S then 1 else 0) := by |
| intro e he |
| have hc := h e he |
| simp only [Crosses] at hc |
| by_cases hl : G.left e ∈ S <;> by_cases hr : G.right e ∈ S <;> |
| simp_all |
| have hh := G.matching_incidence_count M S |
| have heqsum := Finset.sum_congr rfl heq |
| rw [heqsum, ← Finset.mul_sum] at hh |
| obtain ⟨k, hk⟩ := hS |
| omega |
| |
| |
| omit [Fintype V] [DecidableEq E] in |
| theorem degree_linear (G : LooplessGraph V E) (x y : E → ℝ) (a b : ℝ) (v : V) : |
| G.degree (a • x+b • y) v = a*G.degree x v+b*G.degree y v := by |
| have he : ∀ e, (if G.Incident v e then (a • x+b • y) e else 0) = |
| a*(if G.Incident v e then x e else 0)+b*(if G.Incident v e then y e else 0) := by |
| intro e |
| by_cases h : G.Incident v e <;> simp [h] |
| simp only [degree, he, Finset.sum_add_distrib, ← Finset.mul_sum] |
| |
| omit [Fintype V] [DecidableEq E] in |
| theorem cutMass_linear (G : LooplessGraph V E) (x y : E → ℝ) (a b : ℝ) (S : Finset V) : |
| G.cutMass (a • x+b • y) S = a*G.cutMass x S+b*G.cutMass y S := by |
| simp [cutMass, Finset.sum_add_distrib, Finset.mul_sum] |
| |
| omit [Fintype V] [DecidableEq E] in |
| theorem convex_constraintPolytope (G : LooplessGraph V E) : |
| Convex ℝ G.constraintPolytope := by |
| intro x hx y hy a b ha hb hab |
| refine ⟨fun e => add_nonneg (mul_nonneg ha (hx.1 e)) (mul_nonneg hb (hy.1 e)), ?_, ?_⟩ |
| · intro v |
| rw [degree_linear, hx.2.1, hy.2.1, mul_one, mul_one, hab] |
| · intro S hS |
| rw [cutMass_linear] |
| calc |
| 1 = a*1+b*1 := by linarith |
| _ ≤ _ := add_le_add (mul_le_mul_of_nonneg_left (hx.2.2 S hS) ha) |
| (mul_le_mul_of_nonneg_left (hy.2.2 S hS) hb) |
| |
| omit [Fintype V] in |
| theorem indicator_mem_constraintPolytope (G : LooplessGraph V E) (M : G.Matching) : |
| G.indicator M ∈ G.constraintPolytope := by |
| refine ⟨?_, G.degree_indicator M, ?_⟩ |
| · intro e; simp [indicator]; split <;> norm_num |
| · intro S hS |
| obtain ⟨e, he, hc⟩ := G.matching_crosses_odd M S hS |
| have h := Finset.single_le_sum (f := G.indicator M) |
| (s := G.cut S) (a := e) (by intro e' _; simp [indicator]; split <;> norm_num) |
| ((G.mem_cut S e).mpr hc) |
| simpa only [cutMass, indicator, he, ite_true] using h |
| |
| omit [Fintype V] in |
| theorem polytope_subset_constraintPolytope (G : LooplessGraph V E) : |
| G.polytope ⊆ G.constraintPolytope := by |
| apply convexHull_min |
| · rintro x ⟨M, rfl⟩ |
| exact G.indicator_mem_constraintPolytope M |
| · exact G.convex_constraintPolytope |
| |
| omit [Fintype V] [DecidableEq E] in |
| theorem constraint_coordinate_le_one (G : LooplessGraph V E) {x : E → ℝ} |
| (hx : x ∈ G.constraintPolytope) (e : E) : x e ≤ 1 := by |
| have h := Finset.single_le_sum (f := fun e' => if G.Incident (G.left e) e' then x e' else 0) |
| (s := Finset.univ) (a := e) (by |
| intro e' _; split; exact hx.1 e'; rfl) (Finset.mem_univ e) |
| have he : G.Incident (G.left e) e := Or.inl rfl |
| change (if G.Incident (G.left e) e then x e else 0) ≤ G.degree x (G.left e) at h |
| simpa only [he, ite_true, hx.2.1] using h |
| |
| omit [Fintype V] [DecidableEq E] in |
| theorem continuous_degree (G : LooplessGraph V E) (v : V) : |
| Continuous (fun x : E → ℝ => G.degree x v) := by |
| unfold degree |
| apply continuous_finsetSum |
| intro e _ |
| split <;> fun_prop |
| |
| omit [Fintype V] [DecidableEq E] in |
| theorem continuous_cutMass (G : LooplessGraph V E) (S : Finset V) : |
| Continuous (fun x : E → ℝ => G.cutMass x S) := by |
| unfold cutMass |
| fun_prop |
| |
| omit [Fintype V] [DecidableEq E] in |
| theorem isClosed_constraintPolytope (G : LooplessGraph V E) : |
| IsClosed G.constraintPolytope := by |
| simp only [constraintPolytope, Set.ofPred_and, Set.ofPred_forall] |
| exact (isClosed_iInter fun e => isClosed_le continuous_const (continuous_apply e)).inter |
| ((isClosed_iInter fun v => isClosed_eq (G.continuous_degree v) continuous_const).inter |
| (isClosed_iInter fun S => isClosed_iInter fun _ => |
| isClosed_le continuous_const (G.continuous_cutMass S))) |
| |
| omit [Fintype V] [DecidableEq E] in |
| theorem isCompact_constraintPolytope (G : LooplessGraph V E) : |
| IsCompact G.constraintPolytope := by |
| apply (isCompact_Icc : IsCompact (Set.Icc (0 : E → ℝ) 1)).of_isClosed_subset |
| G.isClosed_constraintPolytope |
| intro x hx |
| exact ⟨hx.1, fun e => G.constraint_coordinate_le_one hx e⟩ |
| |
| end LooplessGraph |
| |
| end MatchingEntropy |
| |
| end |
| |
| end OAI |
| |
