Maximal extension in a finite preorder
ProvedWilliamsonShmoys.exists_maximal_extension_of_finite_preorder__p2m_7fc18aad_d07a_49bb_b01c_abff1eddf2b3_a2approximation-algorithmsplanar-graphstreewidth
In a finite preorder, every predicate-satisfying element has a predicate-satisfying extension that is maximal among predicate-satisfying elements above it.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_maximal_extension_of_finite_preorder__p2m_7fc18aad_d07a_49bb_b01c_abff1eddf2b3_a2 {α : Type} [Finite α] [Preorder α] (x : α) (Q : α → Prop) (hx : Q x) : ∃ y : α, Q y ∧ x ≤ y ∧ ∀ z : α, Q z → y ≤ z → z ≤ y := by sorrySource
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).