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Reviewer: Kenneth W. Johnson

The authors designed this book as a text for undergraduate and graduate courses in mathematics, operations research, and computer science departments. The introduction discusses the traveling salesman problem and the matching problem as motivating examples, then considers the problem of measuring the efficiency of an algorithm. Chapter 2, on optimal trees and paths, covers minimal spanning trees and shortest paths. Chapter 3 discusses maximal flow problems, including network flows, applications of maximum flow and minimum cut, push-relabel maximum flow problems, and multicommodity flows. Chapter 4 is about minimum-cost flow problems. Chapter 5 discusses optimal matchings. Chapter 6 covers the integrality of polyhedra. Chapter 7 presents the traveling salesman problem. Chapter 8 discusses matroids. Chapter 9 covers NP and NP-completeness. An appendix briefly treats linear programming, since familiarity with the concepts involved is necessary for an understanding of many of the algorithms. The authors suggest outlines for an introductory course and for a more advanced theoretical course that includes the topics covered in the later chapters. Practical examples are provided, whenever possible, to motivate the definitions, and algorithms are presented that can be used to obtain solutions. The first chapter provides an admirable introduction to the subsequent ideas. The level of the text seems appropriate for well-prepared undergraduates who have been exposed to some combinatorics, and the exposition is clear and spiced with interesting examples. Algorithms are highlighted. Diagrams are easy to follow and take recent improvements into account. The references are comprehensive, but I was surprised to see no mention of Philip Hall in connection with the marriage theorem, and I would have preferred a more elementary statement of the theorem. Numerous exercises of varying difficulty are given after each subsection. The appendix would be difficult for unsophisticated undergraduates with no linear programming experience, and would probably need to be supplemented. The text has many positive qualities and gives a good, clear exposition of the material in the field. I recommend it highly for undergraduates with sufficient maturity and for graduate courses, although I would have reservations about using it to teach a basic undergraduate course to underprepared students.