n-Fold integer programming in cubic time

• Published in: Mathematical programming Vol. 137; no. 1-2; pp. 325 - 341
• Main Authors: Hemmecke, Raymond, Onn, Shmuel, Romanchuk, Lyubov
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.02.2013; Springer Nature B.V
• 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; 68R; 52B; 52C; 90B; 62H; 90C; 68Q
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-011-0490-y

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.