Aggregate local edge bounds across a connected partition

Proved

WilliamsonShmoys.edge_bound_of_connected_vertex_partition__p2m_7d0b953f_c62a_470f_b8ef_9a25b50c3ad5_a4

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Given the stated partition and local bounds on every connected induced subgraph, the graph satisfies the global edge bound when it has at least three vertices. Partition edges by their unique part, use zero edges for singleton parts, at most one for two-vertex parts, and the local estimate for larger parts, then sum and absorb the small-part allowances. This isolates the aggregation argument from the construction of the partition.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.edge_bound_of_connected_vertex_partition__p2m_7d0b953f_c62a_470f_b8ef_9a25b50c3ad5_a4
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hcard : 3 <= Fintype.card V)
    (hlocal : ∀ (s : Finset V),
      (K.induce (↑s : Set V)).Connected →
      (letI : DecidableRel (K.induce (↑s : Set V)).Adj := Classical.decRel _;
       (K.induce (↑s : Set V)).edgeFinset.card <=
         max 1 (3 * Fintype.card s - 6)))
    (parts : Finset (Finset V))
    (hparts :
      (∀ s ∈ parts, s.Nonempty ∧ (K.induce (↑s : Set V)).Connected) ∧
      (∀ v, ∃ s ∈ parts, v ∈ s) ∧
      (∀ s ∈ parts, ∀ t ∈ parts, s ≠ t → Disjoint s t) ∧
      (∀ u v, K.Adj u v → ∃ s ∈ parts, u ∈ s ∧ v ∈ s)) :
    (letI : DecidableRel K.Adj := Classical.decRel K.Adj;
     K.edgeFinset.card <= 3 * Fintype.card V - 6) := by sorry

Source

David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).

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