The book "Geometric Algorithms and Combinatorial Optimization" by Martin Grötschel, László Lovász, and Alexander Schrijver is a comprehensive treatise on the intersection of geometric algorithms and combinatorial optimization. The second corrected edition, published by Springer-Verlag, covers a wide range of topics including combinatorial geometry, the geometry of numbers, and mathematical optimization. The authors discuss two key algorithms: the ellipsoid method and the basis reduction algorithm, which have significant applications in combinatorial optimization. The preface to the second edition highlights the continued relevance of the book despite advancements in research, noting that many new results build upon the models, algorithms, and theorems presented. However, several open problems remain unsolved, such as finding combinatorial polynomial-time algorithms for minimizing submodular functions or finding maximum cliques in perfect graphs. The preface to the first edition emphasizes the historical connection between geometry and optimization, particularly in combinatorial optimization. It introduces the ellipsoid method and the geometry of numbers, detailing their theoretical and practical applications. The book aims to provide a theoretical framework for solving a large number of combinatorial optimization problems in polynomial time, while acknowledging that practical running times may not be optimal. The structure of the book is outlined, with chapters covering mathematical preliminaries, the ellipsoid method, diophantine approximation, and applications to combinatorial optimization. The authors express gratitude to various colleagues and institutions for their contributions and support during the development of the book.The book "Geometric Algorithms and Combinatorial Optimization" by Martin Grötschel, László Lovász, and Alexander Schrijver is a comprehensive treatise on the intersection of geometric algorithms and combinatorial optimization. The second corrected edition, published by Springer-Verlag, covers a wide range of topics including combinatorial geometry, the geometry of numbers, and mathematical optimization. The authors discuss two key algorithms: the ellipsoid method and the basis reduction algorithm, which have significant applications in combinatorial optimization. The preface to the second edition highlights the continued relevance of the book despite advancements in research, noting that many new results build upon the models, algorithms, and theorems presented. However, several open problems remain unsolved, such as finding combinatorial polynomial-time algorithms for minimizing submodular functions or finding maximum cliques in perfect graphs. The preface to the first edition emphasizes the historical connection between geometry and optimization, particularly in combinatorial optimization. It introduces the ellipsoid method and the geometry of numbers, detailing their theoretical and practical applications. The book aims to provide a theoretical framework for solving a large number of combinatorial optimization problems in polynomial time, while acknowledging that practical running times may not be optimal. The structure of the book is outlined, with chapters covering mathematical preliminaries, the ellipsoid method, diophantine approximation, and applications to combinatorial optimization. The authors express gratitude to various colleagues and institutions for their contributions and support during the development of the book.