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Cumulative separator cost over disjoint discarded shores

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

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    Natural Language Statement

    Theorem

    Suppose repeated cuts of an nn-vertex graph discard pairwise disjoint shores BiB_i, and each separator Ai∩BiA_i\cap B_i has cost at most 4ρ∣Ai∣∣Bi∣/γ4\rho|A_i||B_i|/\gamma. Nonnegative costs imply that the union of all these separators has total cost at most 4ρn2/γ4\rho n^2/\gamma. The bound uses ∣Ai∣≤n|A_i|\le n and ∑i∣Bi∣≤n\sum_i|B_i|\le n.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    Lean source view on GitHub

    1import Lax263307.PathSampling
    2
    3/-!
    4---
    5title: Cumulative separator cost over disjoint discarded shores
    6type: theorem
    7---
    8Suppose repeated cuts of an nn-vertex graph discard pairwise disjoint
    9shores BiB_i, and each separator Ai∩BiA_i\cap B_i has cost at most
    104ρ∣Ai∣∣Bi∣/γ4\rho|A_i||B_i|/\gamma. Nonnegative costs imply that the union of all
    11these separators has total cost at most 4ρn2/γ4\rho n^2/\gamma.
    12The bound uses ∣Ai∣≤n|A_i|\le n and ∑i∣Bi∣≤n\sum_i|B_i|\le n.
    13
    14# Attribution
    15
    16Bonnet–Chang, the repeated-cut cost accounting in Lemma 8.
    17-/
    18
    19namespace Lax263307.RepeatedCutAccounting
    20
    21noncomputable def cost {n : ℕ} (x : Fin n → ℝ) (S : Finset (Fin n)) : ℝ := ∑ v ∈ S, x v
    22
    23axiom total_cut_cost {n q : ℕ} (x : Fin n → ℝ) (nonnegative : ∀ v, 0 ≤ x v)
    24 (A B : Fin q → Finset (Fin n)) (disjoint : Pairwise fun i j => Disjoint (B i) (B j))
    25 (ρ γ : ℝ) (rho_nonnegative : 0 ≤ ρ) (gamma_positive : 0 < γ)
    26 (each_cut : ∀ i, cost x (A i ∩ B i) ≤ 4 * ρ * (A i).card * (B i).card / γ) :
    27 cost x (Finset.univ.biUnion fun i => A i ∩ B i) ≤ 4 * ρ * (n : ℝ) ^ 2 / γ
    28
    29end Lax263307.RepeatedCutAccounting
    30
    Show Proof

    Attribution

    Bonnet–Chang, the repeated-cut cost accounting in Lemma 8.

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