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Theory of Linear and Integer Programming. Wiley Series in Discrete
Mathematics & Optimization
Description:
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
Contents:
Introduction and Preliminaries.
Problems, Algorithms, and Complexity.
LINEAR ALGEBRA.
Linear Algebra and Complexity.
LATTICES AND LINEAR DIOPHANTINE EQUATIONS.
Theory of Lattices and Linear Diophantine Equations.
Algorithms for Linear Diophantine Equations.
Diophantine Approximation and Basis Reduction.
POLYHEDRA, LINEAR INEQUALITIES, AND LINEAR PROGRAMMING.
Fundamental Concepts and Results on Polyhedra, Linear Inequalities, and Linear Programming.
The Structure of Polyhedra.
Polarity, and Blocking and Anti-Blocking Polyhedra.
Sizes and the Theoretical Complexity of Linear Inequalities and Linear Programming.
The Simplex Method.
Primal-Dual, Elimination, and Relaxation Methods.
Khachiyan's Method for Linear Programming.
The Ellipsoid Method for Polyhedra More Generally.
Further Polynomiality Results in Linear Programming.
INTEGER LINEAR PROGRAMMING.
Introduction to Integer Linear Programming.
Estimates in Integer Linear Programming.
The Complexity of Integer Linear Programming.
Totally Unimodular Matrices: Fundamental Properties and Examples.
Recognizing Total Unimodularity.
Further Theory Related to Total Unimodularity.
Integral Polyhedra and Total Dual Integrality.
Cutting Planes.
Further Methods in Integer Linear Programming.
References.
Indexes.
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