N-Fold integer programming in cubic time

Raymond Hemmecke, Shmuel Onn, Lyubov Romanchuk

Research output: Contribution to journalArticlepeer-review

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 (ng(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 languageEnglish
Pages (from-to)325-341
Number of pages17
JournalMathematical Programming
Volume137
Issue number1-2
DOIs
StatePublished - Feb 2013

Keywords

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

All Science Journal Classification (ASJC) codes

  • Software
  • General Mathematics

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