# 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.

## 33 Citations

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

- Computer Science
- 2012

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

- Computer Science
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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…

### COMPLEXITY OF SHORT GENERATING FUNCTIONS

- MathematicsForum of Mathematics, Sigma
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It is proved that truncated theta functions are hard for this class of short generating functions, in the sense that these operations can increase the bit length of coefficients of generating functions by a super-polynomial factor.

### BASIS REDUCTION METHODS

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We review lattice based methods to solve integer programming feasibility problems, in particular, the algorithms of Lenstra, and Kannan, and the reformulation methods of Aardal, et al. and of…

### Complexity of optimizing over the integers

- Computer ScienceMathematical Programming
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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.

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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.

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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…

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- Computer Science, MathematicsMath. Program.
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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.

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- MathematicsComputational Complexity Conference
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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.

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