## Abstract

n-Fold integer programming is a fundamental problem with a variety of natural applications in operations research and statistics. Moreover, it is universal and provides a new, variable-dimension, parametrization of all of integer programming. The fastest algorithm for n-fold integer programming predating the present article runs in time O (n^{g(A)}L with L the binary length of the numerical part of the input and g(A) the so-called Graver complexity of the bimatrix A defining the system. In this article we provide a drastic improvement and establish an algorithm which runs in time O (n ^{3} L) having cubic dependency on n regardless of the bimatrix A. Our algorithm works for separable convex piecewise affine objectives as well. Moreover, it can be used to define a hierarchy of approximations for any integer programming problem.

Original language | English |
---|---|

Pages (from-to) | 325-341 |

Number of pages | 17 |

Journal | Mathematical Programming |

Volume | 137 |

Issue number | 1-2 |

DOIs | |

State | Published - Feb 2013 |

## Keywords

- 52C
- 62H
- 68Q
- 68R
- 90B
- 90C
- Mathematics Subject Classification (2000): 52B