n-Fold integer programming in cubic time

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 pr...

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Bibliographic Details
Published in:Mathematical programming 2013-02, Vol.137 (1-2), p.325-341
Main Authors: Hemmecke, Raymond, Onn, Shmuel, Romanchuk, Lyubov
Format: Article
Language:English
Subjects:
Algorithms
Analysis
Approximation
Calculus of Variations and Optimal Control
Optimization
Combinatorics
Full Length Paper
Hierarchies
Integer programming
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Operations research
Parametrization
Statistics
Studies
Theoretical
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Summary: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 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.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-011-0490-y