## Theory of Linear and Integer ProgrammingTheory of Linear and Integer Programming Alexander Schrijver Centrum voor Wiskunde en Informatica, Amsterdam, The Netherlands This book describes the theory of linear and integer programming and surveys the algorithms for linear and integer programming problems, focusing on complexity analysis. It aims at complementing the more practically oriented books in this field. A special feature is the author's coverage of important recent developments in linear and integer programming. Applications to combinatorial optimization are given, and the author also includes extensive historical surveys and bibliographies. The book is intended for graduate students and researchers in operations research, mathematics and computer science. It will also be of interest to mathematical historians. Contents 1 Introduction and preliminaries; 2 Problems, algorithms, and complexity; 3 Linear algebra and complexity; 4 Theory of lattices and linear diophantine equations; 5 Algorithms for linear diophantine equations; 6 Diophantine approximation and basis reduction; 7 Fundamental concepts and results on polyhedra, linear inequalities, and linear programming; 8 The structure of polyhedra; 9 Polarity, and blocking and anti-blocking polyhedra; 10 Sizes and the theoretical complexity of linear inequalities and linear programming; 11 The simplex method; 12 Primal-dual, elimination, and relaxation methods; 13 Khachiyan's method for linear programming; 14 The ellipsoid method for polyhedra more generally; 15 Further polynomiality results in linear programming; 16 Introduction to integer linear programming; 17 Estimates in integer linear programming; 18 The complexity of integer linear programming; 19 Totally unimodular matrices: fundamental properties and examples; 20 Recognizing total unimodularity; 21 Further theory related to total unimodularity; 22 Integral polyhedra and total dual integrality; 23 Cutting planes; 24 Further methods in integer linear programming; Historical and further notes on integer linear programming; References; Notation index; Author index; Subject index |

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### Contents

Introduction and preliminaries | 1 |

Problems algorithms and complexity | 14 |

LINEAR ALGEBRA | 25 |

Notes on linear algebra | 38 |

Theory of lattices and linear diophantine equations | 45 |

Algorithms for linear diophantine equations | 52 |

Diophantine approximation and basis reduction | 60 |

Notes on lattices and linear diophantine equations | 76 |

Further polynomiality results in linear programming | 190 |

Notes on polyhedra linear inequalities and linear | 209 |

Estimates in integer linear programming | 237 |

The complexity of integer linear programming | 245 |

fundamental properties and examples | 266 |

Recognizing total unimodularity | 282 |

Further theory related to total unimodularity | 294 |

Integral polyhedra and total dual integrality | 309 |

POLYHEDRA LINEAR INEQUALITIES | 83 |

The structure of polyhedra | 99 |

Polarity and blocking and antiblocking polyhedra | 112 |

Sizes and the theoretical complexity of linear inequalities | 120 |

The simplex method | 129 |

Primaldual elimination and relaxation methods | 151 |

Khachiyans method for linear programming | 163 |

The ellipsoid method for polyhedra more generally | 172 |

Cutting planes | 339 |

Further methods in integer linear progamming | 360 |

Historical and further notes on integer linear programming | 375 |

381 | |

452 | |

465 | |

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### Common terms and phrases

apply Ax^b box-TDI characterization Chvatal column vector combinatorial components cone constraints continued fraction convex Corollary corresponding cutting plane defined denotes described determinant directed graph ellipsoid method entries equivalent Euclidean algorithm exists facet Farkas feasible solution finite follows Fourier-Motzkin elimination full-dimensional Gaussian elimination given rational Hence Hermite normal form Hilbert basis implies induction integer linear programming integral optimum solution integral solution integral vector iteration knapsack problem lattice lemma linear diophantine equations linear equations linear inequalities linear programming problem linearly independent LP-problem matroid max cx\Ax maximum minimal face minimum Moreover natural number network matrix nonsingular nonzero optimization pivot polyhedra polyhedron polynomial algorithm polynomially bounded polynomially solvable polytope Proof rational matrix rational polyhedron rational vector row vector satisfies Section separation algorithm simplex method solving submatrix Suppose system of linear tableau Theorem total dual integrality totally unimodular unimodular matrices variables vertex vertices x\Ax