Efficient Graph Coloring with Neural Networks: A Physics-Inspired Approach for Large Graphs

TL;DR

This paper presents a physics-inspired neural network framework that efficiently solves large-scale graph coloring problems by combining graph neural networks with principles from statistical mechanics, achieving near-optimal performance near phase transition thresholds.

Contribution

Introduces a novel neural approach integrating physics concepts for scalable graph coloring, effectively operating near fundamental phase boundaries.

Findings

Achieves algorithmic thresholds close to the theoretical dynamical transition.

Generalizes from small to large graphs, maintaining effectiveness.

Reaches near-optimal detection in planted inference regimes.

Abstract

Combinatorial optimization problems near algorithmic phase transitions represent a fundamental challenge for both classical algorithms and machine learning approaches. Among them, graph coloring stands as a prototypical constraint satisfaction problem exhibiting sharp dynamical and satisfiability thresholds. Here we introduce a physics-inspired neural framework that learns to solve large-scale graph coloring instances by combining graph neural networks with statistical-mechanics principles. Our approach integrates a planting-based supervised signal, symmetry-breaking regularization, and iterative noise-annealed neural dynamics to navigate clustered solution landscapes. When the number of iterations scales quadratically with graph size, the learned solver reaches algorithmic thresholds close to the theoretical dynamical transition in random graphs and achieves near-optimal detection performance in the planted inference regime. The model generalizes from small training graphs to instances orders of magnitude larger, demonstrating that neural architectures can learn scalable algorithmic strategies that remain effective in hard connectivity regions. These results establish a general paradigm for learning neural solvers that operate near fundamental phase boundaries in combinatorial optimization and inference.