You Only Randomize Once: Shaping Statistical Properties in Constraint-based PCG

The You-Only-Randomize-Once (YORO) technique provides a robust framework for procedural content generation that bridges the gap between rigid constraint satisfaction and flexible statistical sampling. By pre-processing decision variable orderings before execution, this method enables standard SAT solvers to produce outputs that adhere to specific target distributions while simultaneously satisfying complex, global hard constraints. This approach effectively decouples the statistical shaping of content from the underlying solver logic, allowing developers to influence the aesthetic and structural properties of generated assets without modifying the core solver architecture.

The core mechanism relies on the Gumbel-max trick to generate a randomized variable ordering that mirrors desired tile frequencies. By concentrating randomness into a pre-sampled matrix, the technique forces the solver to prioritize specific outcomes during the tie-breaking process. Experimental validation confirms that this method significantly outperforms uniform or heuristic-based ordering, successfully aligning output distributions with target design goals. Because the technique operates by manipulating variable order rather than the solver’s internal algorithms, it remains highly adaptable across various SAT-solving environments, including those requiring manual configuration or formula transformation.

This methodology represents a significant advancement for game industry workflows, offering a scalable alternative to existing procedural generation tools like WaveFunctionCollapse. By integrating established computational techniques from constraint satisfaction and statistical sampling, the approach ensures that generated content maintains both local coherence and global structural integrity. Ultimately, the findings demonstrate that precise control over variable ordering is a powerful, solver-agnostic lever for shaping the statistical properties of procedurally generated environments, providing a versatile toolset for modern game design.