Diffusion models have shown promise in learning to solve constraint optimization problems. However, they are mostly restricted to problems with binary variables and rely on graph neural networks, hindering their application to a broader range of problems such as those with general discrete variables or constraint structures that necessitate global rather than local reasoning. We investigate the use of Diffusion Transformers to address the aforementioned limitations. A naive implementation performs poorly due to a fundamental mismatch between the standard diffusion process and constraint solving: while the former applies small, incremental denoising across all variables, the latter requires substantially altering specific subsets of variables to attain feasibility or optimality. Our method, Blocked Gibbs Diffusion Transformer (BloGDiT), is the first to address this limitation by replacing standard joint Gaussian denoising with blocked Gaussian denoising. BloGDiT uses iterative block resampling and anneals the block size over time to facilitate large, targeted edits within a block of variables. Across Sudoku, Graph Coloring, Maximum Independent Set, and MaxCut, BloGDiT matches or outperforms existing methods, demonstrating that blocked Gibbs-style diffusion provides a highly effective inductive bias for Transformer-based constraint satisfaction and optimization.
Diffusion-based neural solvers have shown strong promise for combinatorial optimization (CO), but existing methods typically rely on supervised training with large collections of near-optimal solutions. In this work, we extend adjoint-based trajectory optimization methods to discrete combinatorial domains. We formulate diffusion-based CO as a stochastic control problem over Continuous-Time Markov Chains and introduce discrete adjoint dynamics for propagating optimization signals through discrete generative trajectories. Building on this formulation, we propose Combinatorial Adjoint Matching (CAM), an unsupervised training framework for discrete diffusion solvers with structured and low-variance trajectory-level optimization signals. Empirically, CAM consistently outperforms existing unsupervised diffusion baselines and achieves performance competitive with strong supervised diffusion solvers and even traditional solvers across diverse combinatorial optimization problems. Our code is available at https://github.com/Shengyu-Feng/CAM.
This paper proposes a novel learning-based approach to approximately solve instances of mixed-integer optimization problems. These problems are computationally challenging, as they require jointly determining discrete and continuous decisions while satisfying complex combinatorial constraints. The proposed method relies on a graph-based generative diffusion model that learns the discrete component of mixed-integer optimization problems while integrating a training-free feasibility projection operator directly into the reverse diffusion process to steer intermediate samples toward the feasible set throughout generation. Once the discrete decisions are generated, the remaining optimization reduces to a continuous problem that can be solved efficiently (relative to the original problem) using existing numerical methods. The resulting framework named Constrained Graph Diffusion (CGD), is problem-agnostic and can accommodate a broad class of mixed-integer optimization problems through suitable projection operators. We evaluate CGD on optimal transmission switching for ACOPF and discrete portfolio optimization, demonstrating substantial improvements in feasibility and solution quality over learning-based baselines while achieving speedups of up to 425× over state-of-the-art numerical solvers for MINLPs.
Vincenzo Di Vito, Mehdi Taghizadeh, Deepjyoti Deka +2
Sudoku is a representative constraint satisfaction problem that requires global structural reasoning under strict discrete constraints. The existing works of solving Sudoku mainly focus on two dominant approaches, i.e., traditional heuristic and deep learning solver. However, they suffer from two complementary limitations: learning-based solvers lack hard correctness guarantees, while complete symbolic solvers are still prone to long-tail search. To address these shortcomings, we propose a novel diffusion model-guided approach, termed as DiBS, for the branch selection search process. Specifically, DiBS keeps the symbolic solver complete and uses the diffusion model as a branch-ordering guide. The core method is ranking candidate values under the current partial assignment and lightweight consistency signal. Furthermore, we provide an in-depth theoretical proof to reveal how it works and why it works. Experiments on the challenging Royle 17-clue Sudoku benchmark show that our DiBS substantially reduces search cost relative to strong heuristic baselines, especially in nodes, backtracks, and long-tail percentiles. Besides, these results confirm that learned global guidance is effective on hard instances where branch-order mistakes are most expensive. All codes are available at https://github.com/shanxierdan/DiBS.