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Lax18.EquitableCleanup

Equitable cleanup of a bounded partition

concepts/Lax18/EquitableCleanup.lean · lax-18

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    Theorem

    For every error tolerance η > 0 and bound r on the number of classes, there are uniform bounds K,N with the following property. Every partition of a graph on at least N vertices into at most r classes can be replaced by an equitable partition into between its original number of classes and K classes, while losing at most η of its energy.

    Conceptually, each old class is cut into nearly equal small pieces and the few remainders are redistributed. This is the cleanup step that allows the energy-increment proof to maintain the clean textbook formulation without an exceptional class.

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    1import Lax18.PartitionEnergy
    2
    3/-!
    4---
    5title: Equitable cleanup of a bounded partition
    6type: theorem
    7---
    8For every error tolerance η > 0 and bound r on the number of classes,
    9there are uniform bounds K,N with the following property. Every partition
    10of a graph on at least N vertices into at most r classes can be
    11replaced by an equitable partition into between its original number of
    12classes and K classes, while losing at most η of its energy.
    13
    14Conceptually, each old class is cut into nearly equal small pieces and the
    15few remainders are redistributed. This is the cleanup step that allows the
    16energy-increment proof to maintain the clean textbook formulation without an
    17exceptional class.
    18-/
    19
    20namespace Lax18.EquitableCleanup
    21
    22open Lax18.FiniteGraphPartitions
    23open Lax18.PartitionEnergy
    24
    25universe u
    26
    27/-- A bounded partition can be made equitable with arbitrarily small loss of
    28energy, uniformly over all sufficiently large finite graphs. -/
    29axiom equitable_cleanup :
    30 ∀ (η : ℝ) (r : ℕ),
    31 0 < η →
    32 0 < r →
    33 ∃ K N : ℕ,
    34 r ≤ K ∧
    35 r ≤ N ∧
    36 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    37 (G : SimpleGraph V) (P : VertexPartition V),
    38 P.partCount ≤ r →
    39 N ≤ Fintype.card V →
    40 ∃ Q : VertexPartition V,
    41 Q.Equitable ∧
    42 P.partCount ≤ Q.partCount ∧
    43 Q.partCount ≤ K ∧
    44 partitionEnergy G P ≤
    45 partitionEnergy G Q + η
    46
    47end Lax18.EquitableCleanup
    48
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