Node Classification and Search on the Rubik's Cube Graph with GNNs
Alessandro Barro
TL;DR
The paper addresses solving the 3x3x3 Rubik's Cube by learning a geometry-aware heuristic on the cube's Cayley graph, reframing distance estimation as a node classification problem solved with Graph Neural Networks (GNNs). It derives an $A^*$ heuristic from predicted distance-class labels $Y_g$, trained on random-walk generated subgraphs to minimize human input, using the relative distance $d_0(g)=d(g_0,g)$ with diameter $M=26$. Empirically, the GNN-based heuristic can produce shorter solution paths than the DeepCubeA baseline but requires substantially longer search times and expands more nodes, illustrating a trade-off between solution quality and efficiency. The work highlights potential benefits of reducing manual prior knowledge and points to future directions in data design, symmetry exploitation, scalability, and applying the approach to other Cayley-graph puzzles.
Abstract
This study focuses on the application of deep geometric models to solve the 3x3x3 Rubik's Cube. We begin by discussing the cube's graph representation and defining distance as the model's optimization objective. The distance approximation task is reformulated as a node classification problem, effectively addressed using Graph Neural Networks (GNNs). After training the model on a random subgraph, the predicted classes are used to construct a heuristic for $A^*$ search. We conclude with experiments comparing our heuristic to that of the DeepCubeA model.