Theory of Linear and Integer Programming

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John Wiley & Sons, Jul 7, 1998 - Mathematics - 484 pages
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Theory 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
References
381
Notation index
452
Subject index
465
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About the author (1998)

Professor Schrijver has held tenured positions with the Mathematisch Centrum in Amsterdam, and the University of Amsterdam. He has spent leaves of absence in Oxford and Szeged (Hungary). In 1983 he was appointed to the post of Professor of Mathematics at Tilburg University, The Netherlands, with a partial engagement at the Centrum voor Wiskunde en Informatica in Amsterdam.

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