A nonempty finite predicate has a maximum-score member

Proved

WilliamsonShmoys.exists_max_score_member__p2m_456d6f91_af0e_46c2_8bb6_0425e8955342_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite type, any predicate with at least one satisfying element has a satisfying element whose natural-number score is at least that of every other satisfying element.

Preamble

import Mathlib

Formal statement

theorem WilliamsonShmoys.exists_max_score_member__p2m_456d6f91_af0e_46c2_8bb6_0425e8955342_a1
    {α : Type} [Fintype α]
    (P : α → Prop) (score : α → Nat) (a : α) (ha : P a) :
    ∃ b, P b ∧ ∀ c, P c → score c ≤ score b := 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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