Heatmap-based solvers have emerged as a promising paradigm for Combinatorial Optimization (CO). However, we argue that the dominant Supervised Learning (SL) training paradigm suffers from a fundamental objective mismatch: minimizing imitation loss (e.g., cross-entropy) does not guarantee solution cost minimization. We dissect this mismatch into two deficiencies: Decoder-Blindness (being oblivious to the non-differentiable decoding process) and Cost-Blindness (prioritizing structural imitation over solution quality). We empirically demonstrate that these intrinsic flaws impose a hard performance ceiling. To overcome this limitation, we propose CADO (Cost-Aware Diffusion models for Optimization), a streamlined Reinforcement Learning fine-tuning framework that formulates the diffusion denoising process as an MDP to directly optimize the post-decoded solution cost. We introduce Label-Centered Reward, which repurposes ground-truth labels as unbiased baselines rather than imitation targets, and Hybrid Fine-Tuning for parameter-efficient adaptation. CADO achieves state-of-the-art performance across diverse benchmarks, validating that objective alignment is essential for unlocking the full potential of heatmap-based solvers.

Neural network-based Combinatorial Optimization (CO) methods have shown promising results in solving various NP-complete (NPC) problems without relying on hand-crafted domain knowledge. This paper broadens the current scope of neural solvers for NPC problems by introducing a new graph-based diffusion framework, namely DIFUSCO. It formulates NPC problems into a discrete {0, 1}-vector space and uses graph-based denoising diffusion models to generate high-quality solutions. Specifically, we explore diffusion models with Gaussian and Bernoulli noise, respectively, and also introduce an effective inference schedule to improve the generation quality. We evaluate our methods on two well-studied combinatorial optimization problems: Traveling Salesman Problem (TSP) and Maximal Independent Set (MIS). Experimental results show that DIFUSCO strongly outperforms the previous state-of-the-art neural solvers, improving the performance gap between ground-truth and neural solvers from 1.76% to 0.46% on TSP-500, from 2.46% to 1.17% on TSP-1000, and from 3.19% to 2.58% on TSP-10000. For the MIS problem, DIFUSCO outperforms the previous state-of-the-art neural solver on the challenging SATLIB benchmark. Our code is available at this url .

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.

Diffusion models have recently advanced Combinatorial Optimization (CO) as a powerful backbone for neural solvers. However, their iterative sampling process requiring denoising across multiple noise levels incurs substantial overhead. We propose to learn direct mappings from different noise levels to the optimal solution for a given instance, facilitating high-quality generation with minimal shots. This is achieved through an optimization consistency training protocol, which, for a given instance, minimizes the difference among samples originating from varying generative trajectories and time steps relative to the optimal solution. The proposed model enables fast single-step solution generation while retaining the option of multi-step sampling to trade for sampling quality, which offers a more effective and efficient alternative backbone for neural solvers. In addition, within the training-to-testing (T2T) framework, to bridge the gap between training on historical instances and solving new instances, we introduce a novel consistency-based gradient search scheme during the test stage, enabling more effective exploration of the solution space learned during training. It is achieved by updating the latent solution probabilities under objective gradient guidance during the alternation of noise injection and denoising steps. We refer to this model as Fast T2T. Extensive experiments on two popular tasks, the Traveling Salesman Problem (TSP) and Maximal Independent Set (MIS), demonstrate the superiority of Fast T2T regarding both solution quality and efficiency, even outperforming LKH given limited time budgets. Notably, Fast T2T with merely one-step generation and one-step gradient search can mostly outperform the SOTA diffusion-based counterparts that require hundreds of steps, while achieving tens of times speedup.

The Travelling Salesman Problem (TSP) remains a fundamental challenge in combinatorial optimization, inspiring diverse algorithmic strategies. This paper revisits the "heatmap + Monte Carlo Tree Search (MCTS)" paradigm that has recently gained traction for learning-based TSP solutions. Within this framework, heatmaps encode the likelihood of edges forming part of the optimal tour, and MCTS refines this probabilistic guidance to discover optimal solutions. Contemporary approaches have predominantly emphasized the refinement of heatmap generation through sophisticated learning models, inadvertently sidelining the critical role of MCTS. Our extensive empirical analysis reveals two pivotal insights: 1) The configuration of MCTS strategies profoundly influences the solution quality, demanding meticulous tuning to leverage their full potential; 2) Our findings demonstrate that a rudimentary and parameter-free heatmap, derived from the intrinsic k-nearest nature of TSP, can rival or even surpass the performance of complicated heatmaps, with strong generalizability across various scales. Empirical evaluations across various TSP scales underscore the efficacy of our approach, achieving competitive results. These observations challenge the prevailing focus on heatmap sophistication, advocating a reevaluation of the paradigm to harness both components synergistically. The Travelling Salesman Problem (TSP) stands as a quintessential challenge in combinatorial optimization, drawing considerable interest from both theoretical and applied research communities. As a problem characterized by NP-hardness, the TSP has become a benchmark for evaluating the efficacy of novel algorithmic strategies in determining optimal or near-optimal solutions efficiently (Applegate et al., 2009). It has significant practical applications in domains such as logistics, transportation, manufacturing, and telecommunications, where finding efficient routes is crucial for minimizing costs and improving efficiency (Helsgaun, 2017; Nagata & Kobayashi, 2013). Recent advancements in machine learning have inspired a fresh wave of methodologies for tackling the TSP, particularly through the lens of the "heatmap + Monte Carlo Tree Search (MCTS)" paradigm.

Neural network-based Combinatorial Optimization (CO) methods have shown promising results in solving various NP-complete (NPC) problems without relying on hand-crafted domain knowledge. This paper broadens the current scope of neural solvers for NPC problems by introducing a new graph-based diffusion framework, namely DIFUSCO. It formulates NPC problems into a discrete {0, 1}-vector space and uses graph-based denoising diffusion models to generate high-quality solutions. Specifically, we explore diffusion models with Gaussian and Bernoulli noise, respectively, and also introduce an effective inference schedule to improve the generation quality. We evaluate our methods on two well-studied combinatorial optimization problems: Traveling Salesman Problem (TSP) and Maximal Independent Set (MIS). Experimental results show that DIFUSCO strongly outperforms the previous state-of-the-art neural solvers, improving the performance gap between ground-truth and neural solvers from 1.76% to 0.46% on TSP-500, from 2.46% to 1.17% on TSP-1000, and from 3.19% to 2.58% on TSP-10000.