This study explores the integration of Blackout Diffusion into the DIFUSCO framework for combinatorial optimization, specifically targeting the Traveling Salesman Problem (TSP). Inspired by the success of discrete-time diffusion models (D3PM) in maintaining structural integrity, we extend the paradigm to a continuous-time framework, leveraging the unique properties of Blackout Diffusion. Continuous-time modeling introduces smoother transitions and refined control, hypothesizing enhanced solution quality over traditional discrete methods. We propose three key improvements to enhance the diffusion process. First, we transition from a discrete-time-based model to a continuous-time framework, providing a more refined and flexible formulation. Second, we refine the observation time scheduling to ensure a smooth and linear transformation throughout the diffusion process, allowing for a more natural progression of states. Finally, building upon the second improvement, we further enhance the reverse process by introducing finer time slices in regions that are particularly challenging for the model, thereby improving accuracy and stability in the reconstruction phase. Although the experimental results did not exceed the baseline performance, they demonstrate the effectiveness of these methods in balancing simplicity and complexity, offering new insights into diffusion-based combinatorial optimization. This work represents the first application of Blackout Diffusion to combinatorial optimization, providing a foundation for further advancements in this domain. * The code is available for review at https://github.com/Giventicket/BlackoutDIFUSCO.

Typical generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here, we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact (as opposed to variational) analysis. We relate the theory to the existing continuous-state Gaussian diffusion as well as other approaches to discrete diffusion, and identify the corresponding reverse-time stochastic process and score function in the continuous-time setting, and the reverse-time mapping in the discrete-time setting. As an example of this framework, we introduce ``Blackout Diffusion'', which learns to produce samples from an empty image instead of from noise. Numerical experiments on the CIFAR-10, Binarized MNIST, and CelebA datasets confirm the feasibility of our approach. Generalizing from specific (Gaussian) forward processes to discrete-state processes without a variational approximation sheds light on how to interpret diffusion models, which we discuss.