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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

  

Source: Aldous, David J. - Department of Statistics, University of California at Berkeley

 

Collections: Mathematics