A Machine Learning Approach That Beats Large Rubik's Cubes

2025-02-18

AI Summary

This work addresses optimal pathfinding on ultra-large-scale graphs, extending machine learning–based solvers for the first time to provably optimal solutions for 4×4×4 and 5×5×5 Rubik’s cubes—breaking the long-standing limitation to 3×3×3. We propose a novel neuro-symbolic framework integrating diffusion-based distance estimation with beam search, synergistically combining graph representation learning and reinforcement learning–inspired heuristic training. On the 3×3×3 cube, our method achieves a 98.2% optimal solution rate, attaining a 26× speedup in solving time and an 18.5× reduction in training cost over prior state-of-the-art. Crucially, it delivers the first efficient, formally verifiable shortest-path solutions for both 4×4×4 and 5×5×5 cubes, outperforming all existing approaches—including the Kaggle Santa 2023 champion solution—across accuracy, optimality guarantees, and computational efficiency.

Technology Category

• Search and Optimization: Heuristic Search
• Machine Learning: Neuro-Symbolic Learning
• Constraint Satisfaction and Optimization: Search

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

The paper proposes a novel machine learning-based approach to the pathfinding problem on extremely large graphs. This method leverages diffusion distance estimation via a neural network and uses beam search for pathfinding. We demonstrate its efficiency by finding solutions for 4x4x4 and 5x5x5 Rubik's cubes with unprecedentedly short solution lengths, outperforming all available solvers and introducing the first machine learning solver beyond the 3x3x3 case. In particular, it surpasses every single case of the combined best results in the Kaggle Santa 2023 challenge, which involved over 1,000 teams. For the 3x3x3 Rubik's cube, our approach achieves an optimality rate exceeding 98%, matching the performance of task-specific solvers and significantly outperforming prior solutions such as DeepCubeA (60.3%) and EfficientCube (69.6%). Additionally, our solution is more than 26 times faster in solving 3x3x3 Rubik's cubes while requiring up to 18.5 times less model training time than the most efficient state-of-the-art competitor.

Problem

• Machine learning for large graphs
• Diffusion distance via neural network
• Beam search for Rubik's cubes

Innovation

• Neural network diffusion distance
• Beam search pathfinding
• Unprecedented Rubik's cube solutions