Solving a Rubik's Cube Using its Local Graph Structure

Abstract: The Rubix Cube is a 3-dimensional single-player combination puzzle attracting attention in the reinforcement learning community. A Rubix Cube has six faces and twelve possible actions, leading to a small and unconstrained action space and a very large state space with only one goal state. Modeling such a large state space and storing the information of each state requires exceptional computational resources, which makes it challenging to find the shortest solution to a scrambled Rubix cube with limited resources. The Rubix Cube can be represented as a graph, where states of the cube are nodes and actions are edges. Drawing on graph convolutional networks, we design a new heuristic, weighted convolutional distance, for A star search algorithm to find the solution to a scrambled Rubix Cube. This heuristic utilizes the information of neighboring nodes and convolves them with attention-like weights, which creates a deeper search for the shortest path to the solved state.

• The paper introduces a weighted convolutional distance heuristic that integrates graph convolutional networks with the A* search algorithm.
• It shows that the 2-layer approach yields shorter solution lengths and fewer nodes searched compared to DeepCubeA, despite increased computational time.
• The research suggests future optimization of convolution techniques and broader applicability to other combinatorial puzzles.

An Analytical Examination of Solving a Rubik's Cube Using Its Local Graph Structure

This paper presents a novel approach to solving the Rubik's Cube by leveraging its local graph structure and employing graph convolutional networks (GCNs) to enhance the A* search algorithm. Specifically, the authors propose a heuristic termed weighted convolutional distance (WCD), which incorporates information from neighboring nodes through attention-like weights to determine the optimal path to the solved cube state.

Technical Summary

The Rubik's Cube, with its vast state space of approximately $4.325 \times 10^{19}$ possible configurations, presents a formidable challenge in terms of computational efficiency for finding the shortest path to the goal state. Prior methods, such as DeepCubeA, have successfully utilized neural networks alongside the A* search algorithm to efficiently navigate this substantial state space. By conceptualizing the Rubik's Cube as a graph where each configuration is a node and each move is an edge, the authors integrate GCNs for a more informed exploration of possible solutions.

The core contribution is the weighted convolutional distance heuristic, which derives a distance measure based on the probability of selecting each of the cube's twelve actions. This heuristic leverages GCN's message-passing mechanism, facilitating a deeper and more effective search. The formula for WCD adapts through multiple layers, offering a progressively comprehensive understanding of the local graph structure.

Performance Evaluation

The evaluation of the proposed heuristic involves comparing the performance of 1-layer and 2-layer weighted convolutional distance approaches against the DeepCubeA heuristic. The results show significant improvements in the precision of pathfinding: while the 2-layer WCD approach yields slightly shorter solution lengths and substantially reduces the number of nodes explored, it does so at the cost of increased computational time:

• 1-layer WCD: Average solution length of 6.525 moves, average computational time of 51.757 seconds, and 1,097 searched nodes.
• 2-layer WCD: Average solution length of 6.455 moves, extended time of 326.613 seconds, and 535 searched nodes.
• DeepCubeA: Average solution length of 6.875 moves, computational time of 11.467 seconds, and 2,938 searched nodes.

Implications and Future Directions

The findings suggest that the proposed weighted convolutional distance heuristic offers a more precise directionality for the A* search, which translates into fewer nodes being evaluated and thus reduced memory usage. This is especially advantageous for solving Rubik's Cubes starting from configurations farther from the goal state.

However, the increased computational time due to the non-matrix form of convolution remains a notable limitation. Moving forward, optimizing this process by leveraging fast convolution techniques typical in GCNs could significantly enhance performance. Additionally, the methodology holds promise for broader applicability to other combinatorial puzzles with comparable state space structures, such as multi-dimensional Rubik's Cubes or sliding tile puzzles.

In conclusion, this research presents a robust approach to solving the Rubik's Cube by integrating concepts from graph theory and convolutional neural networks. The proposed heuristic not only improves the efficacy of finding optimal solutions but also opens pathways for further exploration in related domains.