Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Graph Colouring Is Hard on Average for Polynomial Calculus and Nullstellensatz

Jonas Conneryd, Susanna F. de Rezende, Jakob Nordström, Shuo Pang, Kilian Risse

TL;DR

This work resolves a long-standing open problem by proving that polynomial calculus and Nullstellensatz require asymptotically linear degree to refute that sparse random $d$-regular graphs and sparse Erdős–Rényi graphs are $3$-colourable, over any field. The authors develop a novel degree-$D$ pseudo-reduction framework and a closure-based locality technique that leverages the local sparsity of random graphs, enabling global degree lower bounds via fast patching of local ideals. Their results yield strongly exponential size lower bounds for these algebraic proof systems through the standard size–degree relationship, and they extend to root-of-unity encodings used in algebraic CSPs and to $k$-colourability settings. The findings significantly advance our understanding of average-case hardness for algebraic proofs in graph colouring, bridging gaps between resolution and algebraic proof complexity and offering tools that may apply to other combinatorial problems.

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 Erdős-Rényi random graphs, are $3$-colourable. Using the known relation between size and degree for polynomial calculus proofs, this implies strongly exponential lower bounds on proof size.

Graph Colouring Is Hard on Average for Polynomial Calculus and Nullstellensatz

TL;DR

This work resolves a long-standing open problem by proving that polynomial calculus and Nullstellensatz require asymptotically linear degree to refute that sparse random -regular graphs and sparse Erdős–Rényi graphs are -colourable, over any field. The authors develop a novel degree- pseudo-reduction framework and a closure-based locality technique that leverages the local sparsity of random graphs, enabling global degree lower bounds via fast patching of local ideals. Their results yield strongly exponential size lower bounds for these algebraic proof systems through the standard size–degree relationship, and they extend to root-of-unity encodings used in algebraic CSPs and to -colourability settings. The findings significantly advance our understanding of average-case hardness for algebraic proofs in graph colouring, bridging gaps between resolution and algebraic proof complexity and offering tools that may apply to other combinatorial problems.

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 Erdős-Rényi 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.

Paper Structure

This paper contains 24 sections, 25 theorems, 34 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Our Contribution
  4. Discussion of Proof Techniques
  5. Outline of This Paper
  6. Preliminaries
  7. Proof Complexity
  8. Graph Colouring and Pseudo-Reductions
  9. Algebra Background
  10. Graph Theory
  11. A Simple Resolution Lower Bound for 4-Colourability on Sparse Graphs
  12. Graph Colouring in CNF and the Resolution Proof System
  13. Resolution Lower Bounds
  14. Polynomial Calculus Lower Bounds: The General Framework
  15. Proof Overview
  16. ...and 9 more sections

Key Result

Theorem 1

For any $d\geq 6$, polynomial calculus (and hence also Nullstellensatz) requires asymptotically almost surely linear degree to refute that random $d$-regular graphs, as well as Erdős-Rényi random graphs, are $3$-colourable. These degree lower bounds hold over any field, and also imply exponential lo

Figures (3)

  • Figure 1: A set $U$ with a 3-hop to the left, a 2-hop on top and a lasso to the right.
  • Figure 2: A depiction of the graphs $B_u$ as defined in the proof of Lemma \ref{['lem:colouring-contraction']}.
  • Figure 3: A depiction of the construction of $U_i$ as defined in the proof of Lemma \ref{['lem:closure-lemma']}.

Theorems & Definitions (64)

  • Theorem 1: informal
  • Definition 2: Pseudo-reduction
  • Lemma 3: Razborov98LowerBound
  • proof
  • Lemma 5
  • Definition 6: Sparsity
  • Lemma 7
  • proof
  • Lemma 8: Sparsity lemma
  • Lemma 9: KPGW10ChromaticCFRR02RandomRegularAN05ChromaticLuczak91Chromatic
  • ...and 54 more