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Projecting an unweighted approximation gives a weighted approximation

Lax263307.WeightedProjection · concepts/Lax263307/WeightedProjection.lean · lax-263307

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    Natural Language Statement

    Theorem

    Let an nn-vertex graph have nonnegative rational weights, with positive maximum weight WW, and let 0<ε<10<\varepsilon<1. Use multiplicities tv=⌊wv/τ⌋t_v=\lfloor w_v/\tau\rfloor, where τ=εW/(2n)\tau=\varepsilon W/(2n). A (1−ε/2)(1-\varepsilon/2)-approximate independent set in the blow-up projects to a (1−ε)(1-\varepsilon)-approximate weighted independent set in the original graph.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    Lean source view on GitHub

    1import Lax263307.RoundingBounds
    2import Lax263307.BlowUpIndependentSets
    3
    4/-!
    5---
    6title: Projecting an unweighted approximation gives a weighted approximation
    7type: theorem
    8---
    9Let an nn-vertex graph have nonnegative rational weights, with positive
    10maximum weight WW, and let 0<ε<10<\varepsilon<1.
    11Use multiplicities tv=⌊wv/τ⌋t_v=\lfloor w_v/\tau\rfloor, where
    12τ=εW/(2n)\tau=\varepsilon W/(2n). A (1−ε/2)(1-\varepsilon/2)-approximate independent
    13set in the blow-up projects to a (1−ε)(1-\varepsilon)-approximate weighted
    14independent set in the original graph.
    15
    16# Attribution
    17
    18Bonnet–Chang, the approximation guarantee in the proof of Theorem 2.
    19
    20# Formalization notes
    21
    22This is the approximation transfer. Preservation of the graph-class
    23separator hypothesis under blow-ups and the algorithm's running time
    24are separate obligations.
    25-/
    26
    27namespace Lax263307.WeightedProjection
    28
    29open WeightedIndependentSet WeightRounding IndependentBlowUp
    30
    31/-- The two half-accuracy losses combine into the requested weighted guarantee. -/
    32axiom approximation_transfer {n : ℕ} (G : SimpleGraph (Fin n)) (w : Fin n → ℚ)
    33 (ε : ℚ) (nonnegative : ∀ v, 0 ≤ w v) (positive_order : 0 < n)
    34 (positive_accuracy : 0 < ε) (accuracy_lt_one : ε < 1)
    35 (positive_maximum : 0 < maximumWeight w)
    36 (K : Finset (Vertex (copies w ε)))
    37 (approximation : IsCardinalityApproximation G (copies w ε) (ε / 2) K) :
    38 IsApproximation G w ε (project K)
    39
    40end Lax263307.WeightedProjection
    41
    Show Proof

    Attribution

    Bonnet–Chang, the approximation guarantee in the proof of Theorem 2.

    Formalization notes

    This is the approximation transfer. Preservation of the graph-class separator hypothesis under blow-ups and the algorithm's running time are separate obligations.

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