The matching problem is central to graph theory and the theory of algorithms. This book provides a comprehensive and straightforward introduction to the basic methods for designing efficient parallel algorithms for graph matching problems. Written for students at the beginning graduate level, the exposition is largely self-contained and example-driven; prerequisites have been kept to a minimum by including relevant background material. The book contains full details of several new techniques and will be of interest to researchers in computer science, operations research, discrete mathematics, and electrical engineering. The main theoretical tools are presented in three independent chapters, devoted to combinatorial tools, probabilistic tools, and algebraic tools. One of the goals of the book is to show how these three approaches can be combined to develop efficient parallel algorithms. The book represents a meeting point of interesting algorithmic techniques and opens up new algebraic and geometric areas.

Many theorems in graph theory give simple characterizations for testing the existence of objects with certain properties, which can be translated into fast parallel algorithms. However, transforming these tests into algorithms for constructing such objects is often a real challenge. In this thesis we develop fast parallel ("NC") algorithms for several such construction problems.

Various problems in computer science are 'hard', that is NP-complete, and so not realistically computable; thus in order to solve them they have to be approximated. This book is a survey of the basic techniques for approximating combinatorial problems using parallel algorithms. Its core is a collection of techniques that can be used to provide parallel approximations for a wide range of problems (for example, flows, coverings, matchings, travelling salesman problems, graphs), but in order to make the book reasonably self-contained, the authors provide an introductory chapter containing the basic definitions and results. A final chapter deals with problems that cannot be approximated, and the book is ended by an appendix that gives a convenient summary of the problems described in the book. This is an up-to-date reference for research workers in the area of algorithms, but it can also be used for graduate courses in the subject.

A module in a graph is like a black box: all the vertices in the module look the same to vertices not in the module. This paper gives the first $NC$ algorithm for finding the modular decomposition of a graph. The algorithm runs in $O$(log $n$) time using $O(n[superscript]{3})$ processors on a CRCW PRAM. This decomposition is used to obtain fast sequential and parallel algorithms for solving graph problems on graphs of bounded module size, e.g. the class of cographs where each module with more than one vertex is either disconnected or its complement is disconnected. These graph problems include minimum coloring, maximum clique, matching, Hamiltonian circuit, and maximum cut. Many of these problems can be solved with $O(n[superscript]{3})$ processors in $O$(log $n$) time. All of them can be solved in $NC$. Our modular decomposition algorithm can be used to obtain more efficient algorithms for recognizing and orienting comparability graphs.

Mathematics of Computing -- Parallelism.