🤖 AI Summary
This work investigates whether standard continuous-time generative models can effectively capture the probability distribution of highly sparse, globally constrained discrete structures—such as Sudoku puzzles. Treating complete Sudoku grids as a discrete subset within a continuously relaxed space, we train flow-matching and score-based generative models, and systematically compare the performance of ODE-, SDE-, and DDPM-style sampling in both unconditional generation and constraint-satisfaction tasks. We demonstrate for the first time that such models can assign non-zero probability mass to globally constrained combinatorial structures and satisfy constraints through stochastic sampling. Empirical results show that score-based stochastic sampling is the most stable, while DDPM-style sampling achieves the highest efficacy. Although less sample-efficient than specialized solvers, our approach establishes the feasibility of using general-purpose probabilistic generative models as Sudoku solvers, thereby extending the applicability of continuous generative modeling to discrete reasoning problems.
📝 Abstract
Can standard continuous-time generative models represent distributions whose support is an extremely sparse, globally constrained discrete set? We study this question using completed Sudoku grids as a controlled testbed, treating them as a subset of a continuous relaxation space. We train flow-matching and score-based models along a Gaussian probability path and compare deterministic (ODE) sampling, stochastic (SDE) sampling, and DDPM-style discretizations derived from the same continuous-time training. Unconditionally, stochastic sampling substantially outperforms deterministic flows; score-based samplers are the most reliable among continuous-time methods, and DDPM-style ancestral sampling achieves the highest validity overall. We further show that the same models can be repurposed for guided generation: by repeatedly sampling completions under clamped clues and stopping when constraints are satisfied, the model acts as a probabilistic Sudoku solver. Although far less sample-efficient than classical solvers and discrete-geometry-aware diffusion methods, these experiments demonstrate that classic diffusion/flow formulations can assign non-zero probability mass to globally constrained combinatorial structures and can be used for constraint satisfaction via stochastic search.