The Objective Method: Probabilistic Combinatorial Optimization Summary: The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence David Aldous and J. Michael Steele 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 A Motivating Example: the Assignment Problem . . . . . . . . . . . . . . . . . . . 3 1.2 A Stalking Horse: the Partial Matching Problem . . . . . . . . . . . . . . . . . . . . 4 1.3 Organization of the Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Geometric Graphs and Local Weak Convergence. . . . . . . . . . . . . . 6 2.1 Geometric Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 G as a Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Local Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 The Standard Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 A Prototype: The Limit of Uniform Random Trees . . . . . . . . . . . . . . . . . . 9 3 Maximal Weight Partial Matching on Random Trees . . . . . . . . . . 12 3.1 Weighted Matchings of Graphs in General . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Our Case: Random Trees with Random Edge Weights . . . . . . . . . . . . . . . 12 3.3 Two Obvious Guesses: One Right, One Wrong . . . . . . . . . . . . . . . . . . . . . 12 3.4 Not Your Grandfather's Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5 A Direct and Intuitive Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Collections: Mathematics