The book "Theory of Linear and Integer Programming" by Alexander Schrijver provides a comprehensive overview of linear and integer programming, covering both theoretical foundations and practical algorithms. The content is divided into several parts: 1. **Introduction and Preliminaries**: This section introduces the basic concepts, notation, and preliminaries from linear algebra, matrix theory, Euclidean geometry, and graph theory. It also discusses problems, algorithms, and complexity, including the classes $\mathcal{P}$, $\mathcal{NP}$, and co-$\mathcal{NP}$. 2. **Linear Algebra**: This part delves into the theory of linear algebra, including the Gaussian elimination method, iterative methods, and their applications in solving linear systems and optimization problems. 3. **Lattices and Linear Diophantine Equations**: Here, the book covers the theory of lattices, the Hermite normal form, unimodular matrices, and algorithms for solving linear Diophantine equations. It also discusses Diophantine approximation and basis reduction methods. 4. **Polyhedra, Linear Inequalities, and Linear Programming**: This section explores fundamental concepts and results on polyhedra, linear inequalities, and linear programming. It includes the simplex method, primal-dual methods, and Khachiyan’s ellipsoid method. The book also discusses the complexity of linear programming and related topics. 5. **Integer Linear Programming**: This part introduces integer linear programming (ILP), including the integer hull of a polyhedron, integral polyhedra, and Hilbert bases. It covers estimates in ILP, the complexity of ILP, and methods for solving ILP, such as cutting planes and branch-and-bound techniques. The book is a valuable resource for researchers, students, and practitioners in the fields of mathematics, computer science, and operations research. It provides a thorough understanding of the theoretical and practical aspects of linear and integer programming.The book "Theory of Linear and Integer Programming" by Alexander Schrijver provides a comprehensive overview of linear and integer programming, covering both theoretical foundations and practical algorithms. The content is divided into several parts: 1. **Introduction and Preliminaries**: This section introduces the basic concepts, notation, and preliminaries from linear algebra, matrix theory, Euclidean geometry, and graph theory. It also discusses problems, algorithms, and complexity, including the classes $\mathcal{P}$, $\mathcal{NP}$, and co-$\mathcal{NP}$. 2. **Linear Algebra**: This part delves into the theory of linear algebra, including the Gaussian elimination method, iterative methods, and their applications in solving linear systems and optimization problems. 3. **Lattices and Linear Diophantine Equations**: Here, the book covers the theory of lattices, the Hermite normal form, unimodular matrices, and algorithms for solving linear Diophantine equations. It also discusses Diophantine approximation and basis reduction methods. 4. **Polyhedra, Linear Inequalities, and Linear Programming**: This section explores fundamental concepts and results on polyhedra, linear inequalities, and linear programming. It includes the simplex method, primal-dual methods, and Khachiyan’s ellipsoid method. The book also discusses the complexity of linear programming and related topics. 5. **Integer Linear Programming**: This part introduces integer linear programming (ILP), including the integer hull of a polyhedron, integral polyhedra, and Hilbert bases. It covers estimates in ILP, the complexity of ILP, and methods for solving ILP, such as cutting planes and branch-and-bound techniques. The book is a valuable resource for researchers, students, and practitioners in the fields of mathematics, computer science, and operations research. It provides a thorough understanding of the theoretical and practical aspects of linear and integer programming.