Probability Theory and Combinatorial Optimization

Table of Contents

Preface
1. First View of Problems and Methods. A first example. Long common subsequences
Subadditivity and expected values
Azuma's inequality and a first application
A second example. The increasing-subsequence problem
Flipping Azuma's inequality
Concentration on rates
Dynamic programming
Kingman's subadditive ergodic theorem
Observations on subadditive subsequences
Additional notes
2. Concentration of Measure and the Classical Theorems. The TSP and quick application of Azuma's inequality
Easy size bounds
Another mean Poissonization
The Beardwood-Halton-Hammersly theorem
Karp's partitioning algorithms
Introduction to space-filling curve heuristic
Asymptotics for the space-filling curve heuristic
Additional notes
3. More General Methods. Subadditive Euclidean functionals
Examples. Good, bad and forthcoming
A general L-(infinity) bound
Simple subadditivity and geometric subadditivity
A concentration inequality
Minimal matching
Two-sided bounds and first consequences
Rooted duals and their applications
Lower bounds and best possibilities
Additional remarks
4. Probability in Greedy Algorithms and Linear Programming. Assignment problem
Simplex method for theoreticians
Dyer-Frieze-McDiarmid inequality
Dealing with integral constraints
Distributional bounds
Back to the future
Additional remarks
5. Distributional Techniques and the Objective Method. Motivation for a method
Searching for a candidate object
Topology for nice sets
Information on the infinite tree
Dénoument
Central limit theory
Conditioning method for independence
Dependency graphs and the CLT
Additional remarks
6. Talagrand's Isoperimetric Theory. Talagrand's isoperimetric theory
Two geometric applications of the isoperimetric inequality
Application to the longest-increasing-subsequence problem
Proof of the isoperimetric problem
Application and comparison in the theory of hereditary sets
Suprema of linear functionals
Tail of the assignment problem
Further applications of Talagrand's isoperimetric inequalities
Final considerations on related work
Bibliography
Index.