Probability Theory and Combinatorial Optimization

This monograph provides an introduction to the state of the art of the probability theory that is most directly applicable to combinatorial optimization. The questions that receive the most attention are those that deal with discrete optimization problems for points in Euclidean space, such as the minimum spanning tree, the traveling-salesman tour, and minimal-length matchings. Still, there are several nongeometric optimization problems that receive full treatment, and these include the problems of the longest common subsequence and the longest increasing subsequence. The philosophy that guides the exposition is that analysis of concrete problems is the most effective way to explain even the most general methods or abstract principles.

Spis treści:
Preface
1. First View of Problems and Methods. A first example. Long common subsequences
Subadditivity and expected values
Azumas inequality and a first application
A second example. The increasing-subsequence problem
Flipping Azumas inequality
Concentration on rates
Dynamic programming
Kingmans subadditive ergodic theorem
Observations on subadditive subsequences
Additional notes
2. Concentration of Measure and the Classical Theorems. The TSP and quick application of Azumas inequality
Easy size bounds
Another mean Poissonization
The Beardwood-Halton-Hammersly theorem
Karps 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. Talagrands Isoperimetric Theory. Talagrands 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 Talagrands isoperimetric inequalities
Final considerations on related work
Bibliography
Index.