A Comparison of Abstraction Heuristics for Rubik's Cube

Clemens Büchner, Patrick Ferber, Jendrik Seipp, Malte Helmert

Keywords: abstractions, abstraction heuristics, Rubik's Cube

TL;DR: We compare Cartesian and merge-and-shrink abstractions to pattern databases, the state-of-the-art heuristic for solving Rubik's Cube optimally.

Abstract: Since its invention in 1974, the Rubik’s Cube puzzle fascinates people of all ages. Its rules are simple: the player gets a scrambled cube and rotates the six faces until each face contains only stickers of one color. Nevertheless, finding a short sequence of rotations to solve the cube is hard. We present the first model of Rubik’s Cube for general problem solvers. To obtain a concise model, we require conditional effects. Furthermore, we extend counterexample-guided Cartesian abstraction refinement (CEGAR) to support factored effect tasks, a class of planning tasks with a specific kind of conditional effects which includes Rubik’s Cube. Finally, we evaluate how newer types of abstraction heuristics compare against pattern database (PDB) heuristics, the state-of-the-art for solving Rubik’s Cube. We find that PDBs still outperform the more general Cartesian and merge-and-shrink abstractions. However, in contrast to PDBs, Cartesian abstractions yield perfect heuristics up to a certain problem difficulty. These findings raise interesting questions for future research.