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Lax17.SpanningTreeRounding

Bounded-degree spanning-tree relaxation

concepts/Lax17/SpanningTreeRounding.lean · lax-17

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    Definition

    A feasible bounded-degree spanning-tree point assigns nonnegative rational weights to the edges, has total weight ∣V∣−1|V|-1, satisfies every forest inequality, and has fractional degree at most BB at every vertex. This is the unweighted relaxation used by the Singh–Lau rounding theorem.

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    1import Mathlib.Data.Rat.Defs
    2import Lax17.Degree
    3
    4/-!
    5---
    6title: Bounded-degree spanning-tree relaxation
    7type: definition
    8---
    9A feasible bounded-degree spanning-tree point assigns nonnegative rational
    10weights to the edges, has total weight \(|V|-1\), satisfies every forest
    11inequality, and has fractional degree at most \(B\) at every vertex. This is
    12the unweighted relaxation used by the Singh--Lau rounding theorem.
    13-/
    14
    15namespace Lax17.SpanningTreeRounding
    16
    17universe u
    18
    19open Finset
    20
    21/-- Both endpoints of an unordered pair lie in `S`. -/
    22def PairInside {V : Type u} [DecidableEq V]
    23 (S : Finset V) (e : Sym2 V) : Prop :=
    24 e.toFinset ⊆ S
    25
    26/-- The finite edge set of `G`. -/
    27noncomputable def edges {V : Type u} [Fintype V]
    28 (G : SimpleGraph V) : Finset (Sym2 V) :=
    29 G.edgeSet.toFinite.toFinset
    30
    31/-- The edges of `G` with both endpoints in `S`. -/
    32noncomputable def internalEdges {V : Type u} [Fintype V] [DecidableEq V]
    33 (G : SimpleGraph V) (S : Finset V) : Finset (Sym2 V) :=
    34 @Finset.filter (Sym2 V) (PairInside S) (Classical.decPred _) (edges G)
    35
    36/-- The edges of `G` incident with `v`. -/
    37noncomputable def incidentEdges {V : Type u} [Fintype V] [DecidableEq V]
    38 (G : SimpleGraph V) (v : V) : Finset (Sym2 V) :=
    39 (edges G).filter fun e => v ∈ e
    40
    41/-- A feasible point of the bounded-degree spanning-tree relaxation. -/
    42structure FeasiblePoint {V : Type u} [Fintype V] [DecidableEq V]
    43 (G : SimpleGraph V) (B : ℕ) where
    44 weight : Sym2 V → ℚ
    45 nonnegative : ∀ e : Sym2 V, e ∈ edges G → 0 ≤ weight e
    46 total :
    47 (edges G).sum weight = (Fintype.card V - 1 : ℕ)
    48 forest :
    49 ∀ S : Finset V, S ≠ Finset.univ →
    50 (internalEdges G S).sum weight ≤ (S.card - 1 : ℕ)
    51 degree :
    52 ∀ v : V, (incidentEdges G v).sum weight ≤ B
    53
    54end Lax17.SpanningTreeRounding
    55

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