Graph Colouring Is Hard on Average for Polynomial Calculus and Nullstellensatz

2023-11-06

IEEE Annual Symposium on Foundations of Computer Science

AI Summary

This work investigates the average-case proof complexity of refuting 3-colorability for sparse random graphs—specifically, random regular graphs and Erdős–Rényi graphs—in Polynomial Calculus (PC) and Nullstellensatz (NS) proof systems. We establish the first linear-degree lower bounds for both graph families: over any field, any PC or NS refutation of 3-colorability requires degree Ω(n). Leveraging degree–size trade-off theorems, this implies strong exponential lower bounds of 2^Ω(n) on proof size. Our results break a long-standing barrier in average-case algebraic proof complexity, demonstrating inherent average-case hardness of 3-colorability in PC/NS. This provides a foundational benchmark for the algebraic proof complexity of random constraint satisfaction problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements

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

Problem

• Prove polynomial calculus requires linear degree for graph coloring refutation
• Establish strongly exponential lower bounds on proof size
• Analyze sparse random regular and Erdős-Rényi graphs' 3-colourability

Innovation

• Proves linear degree requirement for polynomial calculus
• Applies to sparse random regular graphs
• Establishes strongly exponential proof size bounds