# Integer Programming and Algorithmic Geometry of Numbers - A tutorial

```@inproceedings{Eisenbrand2010IntegerPA,
title={Integer Programming and Algorithmic Geometry of Numbers - A tutorial},
author={Friedrich Eisenbrand},
booktitle={50 Years of Integer Programming},
year={2010}
}```
This chapter surveys a selection of results from the interplay of integer programming and the geometry of numbers. Apart from being a survey, the text is also intended as an entry point into the field. I therefore added exercises at the end of each section to invite the reader to delve deeper into the presented material.
31 Citations

### On the Complexity of Nonlinear Mixed-Integer Optimization

This is a survey on the computational complexity of nonlinear mixedinteger optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number

### On the complexity of nonlinear mixed-integer optimization

This is a survey on the computational complexity of nonlinear mixedinteger optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number

### On the number of integer points in translated and expanded polyhedra

• Mathematics
Discret. Comput. Geom.
• 2021
It is proved that the problem of minimizing the number of integer points in parallel translations of a rational convex polytope in \$\mathbb{R}^6\$ is NP-hard and the Ehrhart quasi-polynomials of rational polytopes can have arbitrary fluctuations.

### FPT-Algorithm for Computing the Width of a Simplex Given by a Convex Hull

• Computer Science
Moscow University Computational Mathematics and Cybernetics
• 2019
The problem of computing the width of simplices generated by the convex hull of their integer vertices is considered. An FPT algorithm, in which the parameter is the maximum absolute value of the

### Complexity of optimizing over the integers

• A. Basu
• Computer Science
Mathematical Programming
• 2022
The main merit of this paper is bringing together all of this information under one unifying umbrella with the hope that this will act as yet another catalyst for more interaction across the continuous-discrete divide.

### Iterated Chvátal-Gomory Cuts and the Geometry of Numbers

• Mathematics
SIAM J. Optim.
• 2014
This work provides a partial answer to the strongest cuts of any single CG-cut, by presenting a polynomial-time algorithm that yields an iterate that is strong in a certain well-defined sense.

### Basis reduction and the complexity of branch-and-bound

• Mathematics
SODA '10
• 2010
The classical branch-and-bound algorithm for the integer feasibility problem [EQUATION] has exponential worst case complexity. We prove that it is surprisingly efficient on reformulations of

### An effective branch-and-bound algorithm for convex quadratic integer programming

• Computer Science, Mathematics
Math. Program.
• 2012
A branch-and-bound algorithm for minimizing a convex quadratic objective function over integer variables subject to convex constraints by exploiting the integrality of the variables using suitably-defined lattice-free ellipsoids.

### The computational complexity of integer programming with alternations

• Mathematics
Computational Complexity Conference
• 2017
It is proved that integer programming with three quantifier alternations is \$NP-complete, even for a fixed number of variables, and it is shown that for two polytopes, counting the projection of integer points in \$Q \backslash P\$ is \$\#P\$-complete.