Graph Colouring Is Hard on Average for Polynomial Calculus and Nullstellensatz
Computational Complexity
2025-03-24 v1
Abstract
We prove that polynomial calculus (and hence also Nullstellensatz) over any field requires linear degree to refute that sparse random regular graphs, as well as sparse Erd\H{o}s-R\'{e}nyi random graphs, are -colourable. Using the known relation between size and degree for polynomial calculus proofs, this implies strongly exponential lower bounds on proof size.
Keywords
Cite
@article{arxiv.2503.17022,
title = {Graph Colouring Is Hard on Average for Polynomial Calculus and Nullstellensatz},
author = {Jonas Conneryd and Susanna F. de Rezende and Jakob Nordström and Shuo Pang and Kilian Risse},
journal= {arXiv preprint arXiv:2503.17022},
year = {2025}
}
Comments
An extended abstract appeared in FOCS'23