Huge Unimodular N-Fold Programs

TL;DR

This paper proves that the huge $n$-fold integer programming problem with variable types can be solved in polynomial time, extending previous results that were limited to fixed numbers of types or layers.

Contribution

It establishes the polynomial-time solvability of huge $n$-fold integer programs with a variable number of types, resolving an open problem in the field.

Findings

Polynomial-time algorithm for huge $n$-fold integer programming with variable $t$.

Extension of previous fixed-$t$ results to variable $t$ cases.

Applicability to totally unimodular matrices in large-scale integer programming.

Abstract

Optimization over $l\times m\times n$ integer $3$-way tables with given line-sums is NP-hard already for fixed $l=3$, but is polynomial time solvable with both $l,m$ fixed. In the {\em huge} version of the problem, the variable dimension $n$ is encoded in {\em binary}, with $t$ {\em layer types}. It was recently shown that the huge problem can be solved in polynomial time for fixed $t$, and the complexity of the problem for variable $t$ was raised as an open problem. Here we solve this problem and show that the huge table problem can be solved in polynomial time even when the number $t$ of types is {\em variable}. The complexity of the problem over $4$-way tables with variable $t$ remains open. Our treatment goes through the more general class of {\em huge $n$-fold integer programming problems}. We show that huge integer programs over $n$-fold products of totally unimodular matrices can be solved in polynomial time even when the number $t$ of brick types is variable.