Rank-maximal extension in a finite preorder
ProvedWilliamsonShmoys.exists_rank_maximal_extension__p2m_15a542b2_5d3b_464a_bec6_5f7385b1f62b_a1approximation-algorithmsplanar-graphstreewidth
In a finite preorder, if a natural-valued rank strictly increases along every proper comparable pair, every element satisfying a predicate has a predicate-satisfying extension that is maximal among comparable predicate-satisfying extensions. Choose a maximum-rank candidate among the finite eligible elements.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_rank_maximal_extension__p2m_15a542b2_5d3b_464a_bec6_5f7385b1f62b_a1
{α : Type} [Fintype α] [Preorder α]
(P : α → Prop) (rank : α → Nat)
(hstrict : ∀ a b, a ≤ b → a ≠ b → rank a < rank b)
(a : α) (ha : P a) :
∃ b, P b ∧ a ≤ b ∧ ∀ c, P c → b ≤ c → c ≤ b := 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).