Combine component edge bounds arithmetically

Proved

WilliamsonShmoys.component_arithmetic__p2m_24514a04_46c3_4995_8d84_ff88ef385ed2_a2

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For a finite list of component vertex and edge counts whose totals are n and E, combine the large-component 3n − 6 bounds and small-component edge-at-most-vertex bounds to prove E ≤ 3n − 6 when n ≥ 3.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.component_arithmetic__p2m_24514a04_46c3_4995_8d84_ff88ef385ed2_a2
    (n E : Nat)
    (parts : List (Nat × Nat))
    (hvertices : (parts.map Prod.fst).sum = n)
    (hedges : (parts.map Prod.snd).sum = E)
    (hparts : ∀ p ∈ parts,
      (3 ≤ p.1 → p.2 ≤ 3 * p.1 - 6) ∧
      (p.1 < 3 → p.2 ≤ p.1))
    (hcard : 3 ≤ n) :
    E ≤ 3 * n - 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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