Integer Programming and Algorithmic Geometry of Numbers - A tutorial

  title={Integer Programming and Algorithmic Geometry of Numbers - A tutorial},
  author={Friedrich Eisenbrand},
  booktitle={50 Years of Integer Programming},
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. 

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


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.


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

  • 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.

Convex minization over Z2

Iterated Chvátal-Gomory Cuts and the Geometry of Numbers

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

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

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

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.



A linear algorithm for integer programming in the plane

Abstract.We show that a 2-variable integer program, defined by m constraints involving coefficients with at most φ bits, can be solved with O(m+φ) arithmetic operations on rational numbers of size

The Design and Analysis of Computer Algorithms

This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.

Fast Integer Programming in Fixed Dimension

It is shown that the optimum of an integer program in fixed dimension, which is defined by a fixed number of constraints, can be computed with O(s) basic arithmetic operations, where s is the binary

Integer Programming with a Fixed Number of Variables

It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers.

On integer points in polyhedra: A lower bound

An example showing that PI can have as many as Ω(ϕn−1) vertices is given, which uses the Dirichlet unit theorem.

On integer points in polyhedra

An algorithm which determines the number of integer points in a polyhedron to within a multiplicative factor of 1+ε in time polynomial inm, ϕ and 1/ε when the dimensionn is fixed is described.

Algorithmic theory of numbers, graphs and convexity

  • L. Lovász
  • Mathematics
    CBMS-NSF regional conference series in applied mathematics
  • 1986
How to Round Numbers Preliminaries and some Applications in Combinatorics Cuts and Joins Chromatic Number, Cliques and Perfect Graphs Minimizing a Submodular Function.

Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix

Recently, Frumkin [9] pointed out that none of the well-known algorithms that transform an integer matrix into Smith [16] or Hermite [12] normal form is known to be polynomially bounded in its runn...

Complexity of the Frobenius problem

Consider the Frobenius Problem: Given positive integersa1,...,an withai ≥ 2 and such that their greatest common divisor is one, find the largest natural number that is not expressible as a