N-fold integer programming in cubic time

TL;DR

This paper presents a groundbreaking algorithm for N-fold integer programming that runs in cubic time with respect to the problem size, significantly improving over previous methods and applicable to various extensions.

Contribution

It introduces a cubic-time algorithm for N-fold integer programming, independent of the bimatrix complexity, and extends to convex objectives and variable entries.

Findings

Algorithm runs in O(n^3 L) time, independent of bimatrix complexity.

Applicable to separable convex piecewise affine objectives.

Can be used to create approximation hierarchies for integer programming.

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 can be extended to separable convex piecewise affine objectives as well, and also to systems defined by bimatrices with variable entries. Moreover, it can be used to define a hierarchy of approximations for any integer programming problem.