Can Continuous-Time Diffusion Models Generate and Solve Globally Constrained Discrete Problems? A Study on Sudoku

• URL: http://arxiv.org/abs/2601.20363v1
• Date: Wed, 28 Jan 2026 08:26:54 GMT
• Title: Can Continuous-Time Diffusion Models Generate and Solve Globally Constrained Discrete Problems? A Study on Sudoku
• Authors: Mariia Drozdova,
• Abstract summary: We use completed Sudoku grids as a controlled testbed, treating them as a subset of a continuous relaxation space.<n>We show that score-based samplers are the most reliable among continuous-time methods, and DDPM-style ancestral sampling achieves the highest validity overall.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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