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Reviewer: Janusz S. Kowalik

The book has two introductory sections and four main parts. Part 1 deals with the theory of linear spaces, the theory of systems of linear equations, and the computational complexity of solving these equations. Part 2 considers solving systems of linear equations in integer variables and the problem of diophantine approximations. In Part 3 the following problems are discussed: solving systems of linear inequalities, solving systems of linear equations in non-negative variables, and solving linear programming problems. The key concepts related to these problems are Farkas' lemma and the duality theorem of linear programming. The solution methods include the classic simplex method and two polynomial-time methods due to Khachiyan and Karmarkar. Part 4 presents the following: solving systems of linear diophantine inequalities, solving systems of linear equations in non-negative integer variables, and solving integer linear programming problems. The methods described include cutting planes, the method of Lenstra, branch-and-bound, and Bender decomposition. The book successfully fulfills the chief objective set by the author—to serve as an advanced complement to more practically oriented books. The emphasis is on fundamentals and theory, but the basic solution methodologies are also included. The approach is modern and includes material related to the computational complexity of the presented algorithms. Also, some very recent results are included, for example Lenstra's algorithm for integer linear programming. Unfortunately, there are no exercises that would help the reader to understand and appreciate the theoretical results. Many historical notes and annotated bibliographic comments enhance the value of the book. I would recommend the book to anyone who is seriously interested in linear and integer programming and especially to those who believe that many computer runs are sufficient for understanding the subject.

Reviewer: Saul I. Gass

The author's motivation in writing this very excellent, up-to-date book was to present an integrated discussion of the theoretical aspects of linear and integer programming. He has succeeded very well, indeed. After two introductory chapters on mathematical preliminaries and algorithmic complexity, the book is divided into four parts: Linear Algebra; Lattices and Linear Diophantine Equations; Polyhedra, Linear Inequalities, and Linear Programming; and Integer Linear Programming. Each part consists of a number of related chapters that are chock-full of all and more than you would ever want to know about the topics, including detailed historical notes and references. The book's timeliness is demonstrated by its chapter on Karmarkar's [1] (as well as Khachiyan's [2]) polynomial-time algorithm for linear programming. Computer scientists should be interested in the book for its discussions on algorithmic complexity, especially the material on the complexity of linear programming and the NP-completeness of integer-programming problems with linear constraints, areas that formed the basis of computational complexity. Although no exercises are included, a student working through the book will be exercised enough and will be well rewarded. The book is an ideal one for advanced seminars, and it is certainly a must for all serious researchers in the field.