Solution Space Topology Guides CMTS Search
This work addresses a fundamental issue in search-guided AI for puzzle solving, offering a novel approach that could enhance efficiency in domains like automated reasoning, though it appears incremental as it builds on prior topological methods by shifting focus to solution space structure.
The paper tackled the problem of guiding Monte Carlo Tree Search (MCTS) in puzzle solving by identifying that grid topology is ineffective because it is constant across instances, and instead proposed using solution space topology based on compatibility graphs of valid color assignments under pattern rules, achieving 100% accuracy in automatic pattern detection for 5 types and showing that algebraic connectivity is the dominant feature for improving search.
A fundamental question in search-guided AI: what topology should guide Monte Carlo Tree Search (MCTS) in puzzle solving? Prior work applied topological features to guide MCTS in ARC-style tasks using grid topology -- the Laplacian spectral properties of cell connectivity -- and found no benefit. We identify the root cause: grid topology is constant across all instances. We propose measuring \emph{solution space topology} instead: the structure of valid color assignments constrained by detected pattern rules. We build this via compatibility graphs where nodes are $(cell, color)$ pairs and edges represent compatible assignments under pattern constraints. Our method: (1) detect pattern rules automatically with 100\% accuracy on 5 types, (2) construct compatibility graphs encoding solution space structure, (3) extract topological features (algebraic connectivity, rigidity, color structure) that vary with task difficulty, (4) integrate these features into MCTS node selection via sibling-normalized scores. We provide formal definitions, a rigorous selection formula, and comprehensive ablations showing that algebraic connectivity is the dominant signal. The work demonstrates that topology matters for search -- but only the \emph{right} topology. For puzzle solving, this is solution space structure, not problem space structure.