MCTS for Structural Inference

• The paper presents a novel framework that integrates topological features such as algebraic connectivity, rigidity, and color variance into MCTS to improve search performance.
• It details methods for constructing compatibility graphs and using dynamic topological bonuses to guide node selection in combinatorial and engineering inference tasks.
• Benchmark experiments on ARC-style puzzles and truss optimization validate the approach by demonstrating enhanced success rates and reduced computational overhead.

Monte Carlo Tree Search (MCTS) for structural inference leverages combinatorial search augmented by structural information—either of solution spaces or domain-specific grammars—to guide decision making in high-complexity inference tasks. Recent work has systematically explored the impact of integrating task-specific topological features into the MCTS node-selection policy, demonstrating that topology-sensitized search can outperform baselines in both combinatorial puzzle domains and engineering applications. Notably, advances include the extraction of solution-space topology via compatibility graphs and the deployment of generative grammar rules for complex object construction.

1. Solution-Space Topology and Compatibility Graphs

A key innovation for structural inference with MCTS is the encoding of partial problem states as solution-space compatibility graphs. For a generic $m \times n$ grid CSP (such as ARC-style puzzles), let $s$ denote a partial assignment of colors to cells, $X$ the set of unfilled cell indices, and $K$ the color alphabet size.

Compatibility Graph Definition:

• Nodes: $V_n = \{ (i, k) : i \in X, k \in \{1, ... , K\} \}$, each representing a candidate assignment of color $k$ to cell $i$.
• Edges: An undirected edge exists between $(i,k)$ and $(j,\ell)$ if assigning $i \mapsto k$ and $j \mapsto \ell$ can co-occur in a complete solution compatible with the pattern rules extracted from $s$.
• Weights: $w((i,k),(j,\ell)) \in [0,1]$ is a soft compatibility, with $1$ signifying maximal compatibility.

Adjacency and degree matrices, $A$ and $D$, are constructed as: $$A_{(i,k),(j,\ell)} = 1_{\{[(i,k),(j,\ell)] \in E_n\}} \cdot w((i,k),(j,\ell)),\quad D_{(i,k),(i,k)} = \sum_{(j,\ell)} A_{(i,k),(j,\ell)}$$ From this, the combinatorial Laplacian $L = D - A$ is formed, whose spectrum encodes geometric and constraint-induced structure in the solution space.

2. Extraction and Role of Topological Features

Several topological features have been identified as impactful for structuring MCTS exploration:

• Algebraic Connectivity ($\lambda_2$): The second-smallest eigenvalue of $L$, serving as a global measure of the graph's connectivity. High $\lambda_2$ indicates constricted, tightly constrained solution spaces; low $\lambda_2$ points to fragmented spaces with many degrees of freedom.
• Rigidity Score ($r_i$): For each unfilled cell $i$, compute the marginal mass over colors $p_k \propto \sum_{j,\ell} A_{(i,k),(j,\ell)}$ (normalized to $\sum_k p_k = 1$), then entropy:

$$H_i = -\sum_k p_k \log p_k,\qquad r_i = 1 - \frac{H_i}{\log K}$$

High $r_i$ means the cell is nearly forced; low $r_i$ indicates maximal flexibility.

• Color-Structure Variance ($\sigma_{colors}$): The standard deviation across cells of the count of feasible colors: $$\sigma_{colors} = \mathrm{stdev}_i(|\{k : (i,k) \in V_n,\ \mathrm{deg}>0\}|)$$. This feature measures spatial inhomogeneity in constraint tightness.

Empirically, algebraic connectivity ($\lambda_2$) has been shown to correlate strongly with task difficulty and search outcome, demonstrating discriminative power across pattern types where instance-level solution-structure (not grid topology) is the true source of search complexity.

3. Integration into the MCTS Selection Policy

Standard MCTS utilizes Upper Confidence Bound for Trees (UCT) for node selection, typically involving empirical mean $Q(s')$ and visit counts. Structural inference augments this score by sibling-normalized topological bonuses based on dynamically computed features:

For each child state $s'$ of node $s$ with siblings $S$: $$f(s') = w_\lambda \cdot \lambda_2(L(G_n(s'))) + w_r \cdot \max_{i \in \mathrm{frontier}(s')} r_i + w_\sigma \cdot \sigma_{colors}(s')$$ (where default weights are $w_\lambda=1$, $w_r=1$, $w_\sigma=0.5$), normalized across $S$: $$\tilde{f}(s') = \frac{f(s') - \mu_S}{\sigma_S + \varepsilon}$$ with $\mu_S$, $\sigma_S$ the mean and stddev over siblings, $\varepsilon = 10^{-6}$.

The final selection score: $$\text{Score}(s') = Q(s') + c \sqrt{\frac{\ln N(\mathrm{parent}(s'))}{N(s') + 1}} + \beta \cdot \tilde{f}(s')$$ with $Q(s')$ the mean value, $N(\cdot)$ visit counts, $c$ the UCB constant, and $\beta = 0.5$ the topological bonus weight.

This structural prior steers the tree policy to prefer subtrees reflecting favorable solution-space connectivity and rigidity, improving efficiency in exploring tight or critical regions of the solution manifold.

4. Empirical Benchmarks and Feature Discrimination

Experiments on 48 synthetic $3 \times 3$ ARC-style tasks (five pattern types, five-color alphabet, 100 MCTS iterations per instance, multiple seeds) provide concrete quantitative support:

Method	Success Rate	Nodes Expanded	Relative Overhead
Vanilla MCTS	45% ± 8%	234 ± 42	1.00×
Grid-Topology control	45% ± 8%	234 ± 42	1.06×
$\lambda_2$ (only)	52% ± 7%	198 ± 38	1.17×
Rigidity (only)	49% ± 8%	215 ± 40	1.11×
Full (λ₂+r+σ)	54% ± 7%	187 ± 35	1.22×

Performance gains—up to 9 percentage points in success—are almost entirely attributable to $\lambda_2$, with rigidity and color-variance providing small but consistent improvements.

Feature values further stratify task hardness:

Pattern	λ₂(compat)	λ₂(grid)	max rᵢ	σ₍colors₎
Rot. Symmetry	5.0 ± 0.1	4.1±0.02	0.89	0.12
Color Frequency	2.1 ± 0.3	4.1±0.02	0.42	0.05
Spatial (None)	1.2 ± 0.4	4.1±0.02	0.28	0.02

Grid Laplacian ($\lambda_2 \approx 4.1$) yields no variation across patterns, whereas compatibility graph $\lambda_2$ ranges widely, tracking constraint tightness and solution-space fragmentation.

5. Extensions: Structural Inference in Engineering Design

MCTS structural inference principles have been generalized to explicit structural design, as in truss optimization (Garayalde et al., 10 Jun 2024). In that setting:

• States are partial truss structures represented by node and bar sets.
• Actions consist of grammar-driven valid expansions (triangle-preserving D/T operators).
• Rewards (for completed structures) are negative maximum displacement from a finite-element solver, enforcing stiffness.

MCTS navigates this space by grammar-encoded feasibility and evaluates intermediate and terminal designs using domain-specific solvers. The selection policy applies an exploitation/exploration balance (parameter $\alpha$, e.g., $\alpha\approx0.3$), with empirical evidence showing greatly reduced simulation calls and high-percentile objective values versus Q-learning baselines.

A plausible implication is that solution-space topology and grammar-guided action sets provide complementary structure, allowing generalization to other engineering domains provided a compatible simulator and local ruleset exist.

6. Distinguishing Features, Limitations, and Outlook

Structural inference via MCTS, when equipped with task-relevant topological signals, achieves demonstrable improvements over both vanilla and structurally uninformed baselines for CSPs and engineering design:

• Discriminative Topology: Only solution-space topology, not intrinsic input connectivity, correlates with search difficulty and outcome.
• Model-Free Prior: The approach does not require learning from data, relying instead on fast pattern-rule detection and principled graph construction.
• Scalability: Application to larger domains is constrained by the cost of topological feature computation and combinatorial branch factor; however, methods such as guided rollouts and sibling normalization can mitigate some computational growth.

In summary, integrating dynamically extracted solution-space topological features or generative-grammar constraints into the MCTS search process constitutes an effective, theoretically principled, and empirically validated approach to structural inference across a variety of discrete optimization domains (Mannucci, 3 Nov 2025, Garayalde et al., 10 Jun 2024). Future work could expand on efficient topological feature extraction for larger problem instances and hybridize with learned policies for further speed-ups.