Review of Probabilistic Combinatorics and Its Applications

Abstract

Probablistic arguments have been applied in many areas of combinatorics and theoretical computer science ever since Erdös first used one to prove bounds on Ramsey numbers. Applications range from constructing graphs with properties useful in building communication networks to almost uniform generation of random structures for the purpose of approximately solving intractable counting problems. This book consists of a series of introductory survey articles on topics in probablistic combinatorics and its applications. (The articles are derived from lectures given in one of a series of short courses sponsored by the American Mathematical Society.) The topics covered include random and random-like graphs, discrete isoperimetric inequalities, rapidly mixing Markov chains, and finite Fourier methods. The emphasis throughout the book is on techniques, with sufficient examples to show their usefulness.