Approximate Graph Colouring and the Crystal with a Hollow Shadow

TL;DR

This paper demonstrates that the lift-and-project hierarchy for linear programming and Diophantine equations cannot solve approximate graph colouring, using combinatorial tensor theory for the proof.

Contribution

It introduces a novel limitation of the lift-and-project hierarchy in solving approximate graph colouring problems.

Findings

Lift-and-project hierarchy fails to solve approximate graph colouring

Uses combinatorial tensor theory for proof

Highlights limitations of linear programming approaches

Abstract

We show that approximate graph colouring is not solved by the lift-and-project hierarchy for the combination of linear programming and linear Diophantine equations. The proof is based on combinatorial tensor theory.