Approximate Graph Colouring and Crystals

TL;DR

This paper demonstrates that approximate graph colouring cannot be solved by any level of the affine integer programming hierarchy, using a novel connection to the construction of symmetric crystal tensors.

Contribution

It introduces a new approach linking graph colouring hardness to the existence of symmetric crystal tensors, providing a combinatorial characterization for their construction.

Findings

Approximate graph colouring is not solvable by any level of the AIP hierarchy.

Established a combinatorial framework for constructing symmetric crystal tensors.

Connected tensor construction to graph colouring complexity.

Abstract

We show that approximate graph colouring is not solved by any level of the affine integer programming (AIP) hierarchy. To establish the result, we translate the problem of exhibiting a graph fooling a level of the AIP hierarchy into the problem of constructing a highly symmetric crystal tensor. In order to prove the existence of crystals in arbitrary dimension, we provide a combinatorial characterisation for realisable systems of tensors; i.e., sets of low-dimensional tensors that can be realised as the projections of a single high-dimensional tensor.