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ValuedCSP.lean
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ValuedCSP.lean
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/-
Copyright (c) 2023 Martin Dvorak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Dvorak
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Matrix.Notation
/-!
# General-Valued Constraint Satisfaction Problems
General-Valued CSP is a very broad class of problems in discrete optimization.
General-Valued CSP subsumes Min-Cost-Hom (including 3-SAT for example) and Finite-Valued CSP.
## Main definitions
* `ValuedCSP`: A VCSP template; fixes a domain, a codomain, and allowed cost functions.
* `ValuedCSP.Term`: One summand in a VCSP instance; calls a concrete function from given template.
* `ValuedCSP.Term.evalSolution`: An evaluation of the VCSP term for given solution.
* `ValuedCSP.Instance`: An instance of a VCSP problem over given template.
* `ValuedCSP.Instance.evalSolution`: An evaluation of the VCSP instance for given solution.
* `ValuedCSP.Instance.IsOptimumSolution`: Is given solution a minimum of the VCSP instance?
* `Function.HasMaxCutProperty`: Can given binary function express the Max-Cut problem?
* `FractionalOperation`: Multiset of operations on given domain of the same arity.
* `FractionalOperation.IsSymmetricFractionalPolymorphismFor`: Is given fractional operation a
symmetric fractional polymorphism for given VCSP template?
## References
* [D. A. Cohen, M. C. Cooper, P. Creed, P. G. Jeavons, S. Živný,
*An Algebraic Theory of Complexity for Discrete Optimisation*][cohen2012]
-/
/-- A template for a valued CSP problem over a domain `D` with costs in `C`.
Regarding `C` we want to support `Bool`, `Nat`, `ENat`, `Int`, `Rat`, `NNRat`,
`Real`, `NNReal`, `EReal`, `ENNReal`, and tuples made of any of those types. -/
@[nolint unusedArguments]
abbrev ValuedCSP (D C : Type*) [OrderedAddCommMonoid C] :=
Set (Σ (n : ℕ), (Fin n → D) → C) -- Cost functions `D^n → C` for any `n`
variable {D C : Type*} [OrderedAddCommMonoid C]
/-- A term in a valued CSP instance over the template `Γ`. -/
structure ValuedCSP.Term (Γ : ValuedCSP D C) (ι : Type*) where
/-- Arity of the function -/
n : ℕ
/-- Which cost function is instantiated -/
f : (Fin n → D) → C
/-- The cost function comes from the template -/
inΓ : ⟨n, f⟩ ∈ Γ
/-- Which variables are plugged as arguments to the cost function -/
app : Fin n → ι
/-- Evaluation of a `Γ` term `t` for given solution `x`. -/
def ValuedCSP.Term.evalSolution {Γ : ValuedCSP D C} {ι : Type*}
(t : Γ.Term ι) (x : ι → D) : C :=
t.f (x ∘ t.app)
/-- A valued CSP instance over the template `Γ` with variables indexed by `ι`. -/
abbrev ValuedCSP.Instance (Γ : ValuedCSP D C) (ι : Type*) : Type _ :=
Multiset (Γ.Term ι)
/-- Evaluation of a `Γ` instance `I` for given solution `x`. -/
def ValuedCSP.Instance.evalSolution {Γ : ValuedCSP D C} {ι : Type*}
(I : Γ.Instance ι) (x : ι → D) : C :=
(I.map (·.evalSolution x)).sum
/-- Condition for `x` being an optimum solution (min) to given `Γ` instance `I`. -/
def ValuedCSP.Instance.IsOptimumSolution {Γ : ValuedCSP D C} {ι : Type*}
(I : Γ.Instance ι) (x : ι → D) : Prop :=
∀ y : ι → D, I.evalSolution x ≤ I.evalSolution y
/-- Function `f` has Max-Cut property at labels `a` and `b` when `argmin f` is exactly
`{ ![a, b] , ![b, a] }`. -/
def Function.HasMaxCutPropertyAt (f : (Fin 2 → D) → C) (a b : D) : Prop :=
f ![a, b] = f ![b, a] ∧
∀ x y : D, f ![a, b] ≤ f ![x, y] ∧ (f ![a, b] = f ![x, y] → a = x ∧ b = y ∨ a = y ∧ b = x)
/-- Function `f` has Max-Cut property at some two non-identical labels. -/
def Function.HasMaxCutProperty (f : (Fin 2 → D) → C) : Prop :=
∃ a b : D, a ≠ b ∧ f.HasMaxCutPropertyAt a b
/-- Fractional operation is a finite unordered collection of D^m → D possibly with duplicates. -/
abbrev FractionalOperation (D : Type*) (m : ℕ) : Type _ :=
Multiset ((Fin m → D) → D)
variable {m : ℕ}
/-- Arity of the "output" of the fractional operation. -/
@[simp]
def FractionalOperation.size (ω : FractionalOperation D m) : ℕ :=
Multiset.card.toFun ω
/-- Fractional operation is valid iff nonempty. -/
def FractionalOperation.IsValid (ω : FractionalOperation D m) : Prop :=
ω ≠ ∅
/-- Valid fractional operation contains an operation. -/
lemma FractionalOperation.IsValid.contains {ω : FractionalOperation D m} (valid : ω.IsValid) :
∃ g : (Fin m → D) → D, g ∈ ω :=
Multiset.exists_mem_of_ne_zero valid
/-- Fractional operation applied to a transposed table of values. -/
def FractionalOperation.tt {ι : Type*} (ω : FractionalOperation D m) (x : Fin m → ι → D) :
Multiset (ι → D) :=
ω.map (fun (g : (Fin m → D) → D) (i : ι) => g ((Function.swap x) i))
/-- Cost function admits given fractional operation, i.e., `ω` improves `f` in the `≤` sense. -/
def Function.AdmitsFractional {n : ℕ} (f : (Fin n → D) → C) (ω : FractionalOperation D m) : Prop :=
∀ x : (Fin m → (Fin n → D)),
m • ((ω.tt x).map f).sum ≤ ω.size • Finset.univ.sum (fun i => f (x i))
/-- Fractional operation is a fractional polymorphism for given VCSP template. -/
def FractionalOperation.IsFractionalPolymorphismFor
(ω : FractionalOperation D m) (Γ : ValuedCSP D C) : Prop :=
∀ f ∈ Γ, f.snd.AdmitsFractional ω
/-- Fractional operation is symmetric. -/
def FractionalOperation.IsSymmetric (ω : FractionalOperation D m) : Prop :=
∀ x y : (Fin m → D), List.Perm (List.ofFn x) (List.ofFn y) → ∀ g ∈ ω, g x = g y
/-- Fractional operation is a symmetric fractional polymorphism for given VCSP template. -/
def FractionalOperation.IsSymmetricFractionalPolymorphismFor
(ω : FractionalOperation D m) (Γ : ValuedCSP D C) : Prop :=
ω.IsFractionalPolymorphismFor Γ ∧ ω.IsSymmetric
variable {C : Type*} [OrderedCancelAddCommMonoid C]
lemma Function.HasMaxCutPropertyAt.rows_lt_aux
{f : (Fin 2 → D) → C} {a b : D} (mcf : f.HasMaxCutPropertyAt a b) (hab : a ≠ b)
{ω : FractionalOperation D 2} (symmega : ω.IsSymmetric)
{r : Fin 2 → D} (rin : r ∈ (ω.tt ![![a, b], ![b, a]])) :
f ![a, b] < f r := by
rw [FractionalOperation.tt, Multiset.mem_map] at rin
rw [show r = ![r 0, r 1] by simp [← List.ofFn_inj]]
apply lt_of_le_of_ne (mcf.right (r 0) (r 1)).left
intro equ
have asymm : r 0 ≠ r 1 := by
rcases (mcf.right (r 0) (r 1)).right equ with ⟨ha0, hb1⟩ | ⟨ha1, hb0⟩
· rw [ha0, hb1] at hab
exact hab
· rw [ha1, hb0] at hab
exact hab.symm
apply asymm
obtain ⟨o, in_omega, rfl⟩ := rin
show o (fun j => ![![a, b], ![b, a]] j 0) = o (fun j => ![![a, b], ![b, a]] j 1)
convert symmega ![a, b] ![b, a] (by simp [List.Perm.swap]) o in_omega using 2 <;>
simp [Matrix.const_fin1_eq]
lemma Function.HasMaxCutProperty.forbids_commutativeFractionalPolymorphism
{f : (Fin 2 → D) → C} (mcf : f.HasMaxCutProperty)
{ω : FractionalOperation D 2} (valid : ω.IsValid) (symmega : ω.IsSymmetric) :
¬ f.AdmitsFractional ω := by
intro contr
obtain ⟨a, b, hab, mcfab⟩ := mcf
specialize contr ![![a, b], ![b, a]]
rw [Fin.sum_univ_two', ← mcfab.left, ← two_nsmul] at contr
have sharp :
2 • ((ω.tt ![![a, b], ![b, a]]).map (fun _ => f ![a, b])).sum <
2 • ((ω.tt ![![a, b], ![b, a]]).map f).sum := by
have half_sharp :
((ω.tt ![![a, b], ![b, a]]).map (fun _ => f ![a, b])).sum <
((ω.tt ![![a, b], ![b, a]]).map f).sum := by
apply Multiset.sum_lt_sum
· intro r rin
exact le_of_lt (mcfab.rows_lt_aux hab symmega rin)
· obtain ⟨g, _⟩ := valid.contains
have : (fun i => g ((Function.swap ![![a, b], ![b, a]]) i)) ∈ ω.tt ![![a, b], ![b, a]] := by
simp only [FractionalOperation.tt, Multiset.mem_map]
use g
exact ⟨_, this, mcfab.rows_lt_aux hab symmega this⟩
rw [two_nsmul, two_nsmul]
exact add_lt_add half_sharp half_sharp
have impos : 2 • (ω.map (fun _ => f ![a, b])).sum < ω.size • 2 • f ![a, b] := by
convert lt_of_lt_of_le sharp contr
simp [FractionalOperation.tt, Multiset.map_map]
have rhs_swap : ω.size • 2 • f ![a, b] = 2 • ω.size • f ![a, b] := nsmul_left_comm ..
have distrib : (ω.map (fun _ => f ![a, b])).sum = ω.size • f ![a, b] := by simp
rw [rhs_swap, distrib] at impos
exact ne_of_lt impos rfl