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Reviewer: Michael Donoho Fry

Intended as both a reference and a graduate text, this book is directed toward mathematically mature readers, although the only specific prerequisite is a background in linear programming. A unifying approach to optimization problems is to formulate them like linear programming problems, while restricting some or all of the variables to the integers. This book is an encyclopedic resource for such formulations, as well as for understanding the structure of and solving the resulting integer programming problems. Theoretical foundations, performance analyses, and practical considerations are included. The first of the book's three parts takes advantage of the reader's assumed familiarity with linear programming. The first chapter is devoted to the art of good problem formulation. The types of problems dealt with in the book, standard examples, and an orientation built on linear programming are established here. Included in the following six chapters are a review of linear programming and elementary graph algorithms and self-contained introductory treatments of polyhedral theory, computational complexity, and algorithms in integer linear algebra. Ellipsoid and projective algorithms for linear programming are presented within a context of computational complexity, and polyhedra are used to give a theoretical reduction of integer programming to linear programming. Part 2 begins by establishing general algorithms; improvements are then made for certain classes of problems. The first two chapters deal with the strategy of trying to reduce problems into the domain of linear programming by improving the constraint inequalities. Attention is then turned to the objective function and to sequences of “good” solutions approaching an “optimal” solution. Improvements for special problems, and heuristics for good solutions, are covered in the last chapter. Part 3 treats integer programming problems that have special structures permitting elegant combinatorial solutions. The major topics here are integer polyhedra, matching, and matroids. Considering the concise language used by the authors, the length of the text (719 pages) can be blamed on the extensive use of examples and figures as well as on a relatively complete and in-depth coverage of the subject. It should succeed nicely as a graduate text and has an extensive set of practical and theoretical exercises, but its material could not be covered completely even in a full-year course. At the very least, all students of optimization should be exposed to its first chapter on problem formulation. Where this book succeeds best, though, is as a reference. The organization of the chapters, the author and subject indices, and the complete documentation of the historical development of the subject make it indispensable to those already intimately familiar with the subject it was intended to help unify.