The Huge Multiway Table Problem

TL;DR

This paper investigates the computational complexity of large threeway tables with fixed layer types, showing polynomial-time solvability when layer types are fixed and NP intersect coNP complexity otherwise, using advanced integer programming techniques.

Contribution

It extends the understanding of the huge multiway table problem by analyzing the impact of fixed versus variable layer types within n-fold integer programming.

Findings

Polynomial-time solvability with fixed layer types.

NP intersect coNP complexity with variable layer types.

Application of integer cones and Graver bases techniques.

Abstract

Deciding the existence of an $l\times m\times n$ integer threeway table with given line-sums is NP-complete already for fixed $l=3$, but is in P with both $l,m$ fixed. Here we consider {\em huge} tables, where the variable dimension $n$ is encoded in {\em binary}. Combining recent results on integer cones and Graver bases, we show that if the number of {\em layer types} is fixed, then the problem is in P, whereas if it is variable, then the problem is in NP intersect coNP. Our treatment goes through the more general class of $n$-fold integer programming problems.