# Theory of linear and integer programming

@inproceedings{Schrijver1986TheoryOL, title={Theory of linear and integer programming}, author={Alexander Schrijver}, booktitle={Wiley-Interscience series in discrete mathematics and optimization}, year={1986} }

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…

## 5,701 Citations

### On linear programming and matrix scaling over the algebraic numbers

- Mathematics, Computer Science
- 1997

### New geometric techniques for linear programming and graph partitioning

- Mathematics
- 2006

The first randomized polynomial-time simplex algorithm for linear programming is presented, answering a question that has been open for over fifty years and an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs is proved and shown to be tight, thereby resolving a conjecture of Spielman and Teng.

### Algorithmic and Complexity Results for Cutting Planes Derived from Maximal Lattice-Free Convex Sets

- MathematicsArXiv
- 2011

The key point of this approach is that the conditions are much more explicit and can be tested in a more direct manner, removing the need for a reduction algorithm.

### Linear Programming Stories

- Mathematics
- 2012

The history of polyhedra, linear inequalities, and linear programming has many diverse origins. Polyhedra have been around since the beginning of mathematics in ancient times. It appears that Fourier…

### Integer programming, lattice algorithms, and deterministic volume estimation

- Mathematics, Computer Science
- 2012

A new 2O(n) nn time algorithm is given, which yields the fastest currently known algorithm for IP and improves on the classic works of Lenstra and Kannan, to give a new and tighter proof of the atness theorem.

### Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems

- Computer Science, MathematicsArXiv
- 2015

This work combines methods from partition analysis with methods from polyhedral geometry, and provides an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.

### Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems

- Computer Science, MathematicsAnnals of Combinatorics
- 2017

This work combines methods from partition analysis with methods from polyhedral geometry, and provides an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.

### Convex Discrete Optimization

- Computer Science, MathematicsEncyclopedia of Optimization
- 2009

Using a combination of algebraic and geometric tools, this work provides polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension.

### Computational algebraic analyses for unimodular or Lawrence-type integer programs

- Mathematics
- 2003

There have been a number of studies of the application of computational algebraic approaches to solve integer programming problems, using Gr ̈ obner bases or standard pairs. By connecting these…

### Polyhedral techniques in combinatorial optimization II: applications and computations

- Mathematics, Computer Science
- 1999

The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems because all extreme points of this formulation are integral, which means that the problem can be solved as a linear programming problem.