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

TL;DR

This study investigates whether continuous-time diffusion models can generate and solve globally constrained discrete problems like Sudoku, demonstrating their potential and limitations in representing such structured distributions.

Contribution

The paper shows that continuous-time diffusion models can represent sparse, constrained discrete sets and be adapted for probabilistic constraint satisfaction tasks.

Findings

Stochastic sampling outperforms deterministic flows in validity.

Score-based samplers are most reliable among continuous methods.

Diffusion models can be used for probabilistic constraint satisfaction.

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.