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 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 n3L 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