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Solution Space Topology Guides CMTS Search

Mirco A. Mannucci

TL;DR

This paper reframes search guidance for Monte Carlo Tree Search by shifting focus from problem-space topology to solution-space topology. It introduces a compatibility graph over $(cell, color)$ pairs to capture the structure of valid solutions under detected pattern rules, and defines topological features such as the algebraic connectivity $\lambda_2$ and rigidity $r_i$ to guide search. A sibling-normalized combination of topological features with standard MCTS signals is proposed, and ablations show that $\lambda_2$ dominates the improvement in performance. Empirical validation on synthetic ARC-like tasks demonstrates that topology-informed search yields substantial gains, with a $2.04\times$ average rollout efficiency gain on ARC-1 tasks and up to $6.25\times$ speedups in cases with strong local symmetry, confirming that the right topology materially improves search efficiency in CSP-like puzzle solving. The work emphasizes that topology matters when it is measured on the correct space (the solution space) and suggests directions for extending to higher-level actions and broader CSP domains.

Abstract

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.

Solution Space Topology Guides CMTS Search

TL;DR

This paper reframes search guidance for Monte Carlo Tree Search by shifting focus from problem-space topology to solution-space topology. It introduces a compatibility graph over pairs to capture the structure of valid solutions under detected pattern rules, and defines topological features such as the algebraic connectivity and rigidity to guide search. A sibling-normalized combination of topological features with standard MCTS signals is proposed, and ablations show that dominates the improvement in performance. Empirical validation on synthetic ARC-like tasks demonstrates that topology-informed search yields substantial gains, with a average rollout efficiency gain on ARC-1 tasks and up to speedups in cases with strong local symmetry, confirming that the right topology materially improves search efficiency in CSP-like puzzle solving. The work emphasizes that topology matters when it is measured on the correct space (the solution space) and suggests directions for extending to higher-level actions and broader CSP domains.

Abstract

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 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.

Paper Structure

This paper contains 42 sections, 1 theorem, 4 equations, 3 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Failed Attempt
  3. The Measurement Problem: Grid Topology is Invariant
  4. The Solution: Solution Space Topology
  5. Mathematical Framework
  6. Notation
  7. Compatibility Graph
  8. Laplacian and Algebraic Connectivity
  9. Rigidity
  10. Selection Formula: UCB with Sibling-Normalized Topological Features
  11. Technical Approach
  12. Pattern Detection
  13. Compatibility Graph Construction and Feature Extraction
  14. Experiments
  15. Setup
  16. ...and 27 more sections

Key Result

Lemma 1

Let $G_{m,n}$ denote the $m \times n$ grid graph with 4-neighborhood adjacency. Define its Laplacian as $L = D - A$, where $D$ is the degree matrix and $A$ is the adjacency matrix. For any two ARC-style tasks on the same $G_{m,n}$ that differ only by color constraints on cells, the Laplacian $L$ and

Figures (3)

  • Figure 1: Grid topology vs. compatibility graph. Grid (left) is task-invariant; $G_c$ (right) varies with constraints and guides search.
  • Figure 2: Algebraic connectivity $\lambda_2$ distribution across task states. Higher $\lambda_2$ indicates tighter constraints.
  • Figure 3: Ablation study: success rate across five methods (bars) and relative runtime overhead (inset). Grid Topology (control) validates Lemma \ref{['lem:grid-invariance']}.

Theorems & Definitions (6)

  • Lemma 1: Grid--Laplacian Invariance
  • proof
  • Definition 1: Compatibility Graph
  • Definition 2: Laplacian and Algebraic Connectivity
  • Definition 3: Rigidity Score
  • Definition 4: ARCTransformationGame