Combinatorial optimization problems (COPs) are fundamental to many real-world applications where efficiently producing high-quality solutions is critical. Recent advances in diffusion-based non-autoregressive models have reformulated solving COPs as a generative process, achieving promising results. However, almost all of these methods still suffer from accumulated errors and high inference costs due to the multi-step stochastic denoising process. To address these issues, we propose EFLOCO, an efficient discrete flow matching method for solving COPs, learning structured and deterministic solution trajectories. EFLOCO replaces noise-driven updates with smooth and guided transitions, thereby improves inference stability and quality. Furthermore, we introduce an adaptive time-step scheduler that makes more efforts in critical transition regions, yielding strong performance under few-step constraints. Experiments on standard Traveling Salesman Problems (TSPs) and Asymmetric TSPs (ATSPs) show that our method consistently outperforms both learning-based and heuristic baselines in terms of solution quality and inference speed.

The Vehicle Routing Problem (VRP) is a critical combinatorial optimization problem with wide-reaching real-world applications, particularly in logistics, transportation. While neural network-based VRP solvers have shown impressive results on test instances similar to training data, their performance often degrades when faced with varying scales and unseen distributions, limiting their practical applicability. To overcome these limitations, we introduce DGL (Dynamic Global-Local Information Aggregation), a novel model that combines global and local information to effectively solve VRPs. DGL dynamically adjusts local node selections within a localized range, capturing local invariance across problems of different scales and distributions, thereby enhancing generalization. At the same time, DGL integrates global context into the decision-making process, providing richer information for more informed decisions. Additionally, we propose a replacement-based self-improvement learning framework that leverages data augmentation and random replacement techniques, further enhancing DGL's robustness. Extensive experiments on synthetic datasets, benchmark datasets, and real-world country map instances demonstrate that DGL achieves state-of-the-art performance, particularly in generalizing to large-scale VRPs and real-world scenarios. These results showcase DGL's effectiveness in solving complex, realistic optimization challenges and highlight its potential for practical applications.

Recent advances in neural models have shown considerable promise in solving Traveling Salesman Problems (TSPs) without relying on much hand-crafted engineering. However, while non-autoregressive (NAR) approaches benefit from faster inference through parallelism, they typically deliver solutions of inferior quality compared to autoregressive ones. To enhance the solution quality while maintaining fast inference, we propose DEITSP, a diffusion model with efficient iterations tailored for TSP that operates in a NAR manner. Firstly, we introduce a one-step diffusion model that integrates the controlled discrete noise addition process with self-consistency enhancement, enabling optimal solution prediction through simultaneous denoising of multiple solutions. Secondly, we design a dual-modality graph transformer to bolster the extraction and fusion of features from node and edge modalities, while further accelerating the inference with fewer layers. Thirdly, we develop an efficient iterative strategy that alternates between adding and removing noise to improve exploration compared to previous diffusion methods. Additionally, we devise a scheduling framework to progressively refine the solution space by adjusting noise levels, facilitating a smooth search for optimal solutions. Extensive experiments on real-world and large-scale TSP instances demonstrate that DEITSP performs favorably against existing neural approaches in terms of solution quality, inference latency, and generalization ability.

Multimodal models leverage complementary information across modalities to enrich feature representations. While visual information shows potential in representing structure for some combinatorial optimization problems (COPs), its application to complex scheduling like the Flexible Job Shop Scheduling Problem (FJSP) remains underexplored. Current learning-based FJSP solvers predominantly rely on handcrafted state features. This dependence can lead to inconsistencies and may not fully capture the problem's intricate dynamics. Crucially, these methods overlook visual modalities. Visual representations offer a distinct advantage by inherently capturing the global topological structure and complex resource interactions within the FJSP state. Unlike localized handcrafted features, this holistic, structural view provides a richer foundation for understanding scheduling complexity and making informed decisions. To overcome these limitations by leveraging visual information-known for representing topological structures and providing richer state representations-we introduce the AO-framework. This multimodal feature fusion approach enhances handcrafted state features by integrating insights from visual data. Our core contribution is a novel fusion mechanism utilizing orthogonal projection and local attention. Unlike traditional methods that often rely on simple concatenation of visual data, our method uniquely reduces redundancy by projecting global image-derived features onto local handcrafted features. This process extracts distinct information inherent to the visual modality, significantly improving the quality and complementarity of the resulting state features and enabling more informed scheduling decisions. To our knowledge, the AO-framework represents the first multimodal framework applied to scheduling problems, demonstrating the significant potential of visual information in this domain. Extensive experiments across various FJSP solvers and datasets confirm that our framework yields substantial enhancements in solution quality, decision-making capabilities, and generalization.

The rise of smart manufacturing under Industry 4.0 introduces mass customization and dynamic production, demanding more advanced and flexible scheduling techniques. The flexible job-shop scheduling problem (FJSP) has attracted significant attention due to its complex constraints and strong alignment with real-world production scenarios. Current deep reinforcement learning (DRL)-based approaches to FJSP predominantly employ constructive methods. While effective, they often fall short of reaching (near-)optimal solutions. In contrast, improvement-based methods iteratively explore the neighborhood of initial solutions and are more effective in approaching optimality. However, the flexible machine allocation in FJSP poses significant challenges to the application of this framework, including accurate state representation, effective policy learning, and efficient search strategies. To address these challenges, this paper proposes a $\textbf{M}$emory-enhanced $\textbf{I}$mprovement $\textbf{S}$earch framework with he$\textbf{t}$erogeneous gr$\textbf{a}$ph $\textbf{r}$epresentation—$\textit{MIStar}$. It employs a novel heterogeneous disjunctive graph that explicitly models the operation sequences on machines to accurately represent scheduling solutions. Moreover, a memory-enhanced heterogeneous graph neural network (MHGNN) is designed for feature extraction, leveraging historical trajectories to enhance the decision-making capability of the policy network. Finally, a parallel greedy search strategy is adopted to explore the solution space, enabling superior solutions with fewer iterations. Extensive experiments on synthetic data and public benchmarks demonstrate that $\textit{MIStar}$ significantly outperforms both traditional handcrafted improvement heuristics and state-of-the-art DRL-based constructive methods.

With the widespread adoption of Internet Protocol (IP) communication technology and web-based platforms, cloud manufacturing has become a significant hallmark of Industry 4.0. Integrating graph algorithms into these web-enabled environments is crucial as they facilitate the representation and analysis of complex relationships in manufacturing processes, enabling efficient decision-making and adaptability in dynamic environments. As a key scheduling problem in cloud manufacturing, the flexible job-shop scheduling problem (FJSP) finds extensive applications in real-world scenarios. However, traditional FJSP-solving methods struggle to meet the efficiency and adaptability demands of cloud manufacturing due to generalization issues and excessive computational time, while reinforcement learning-based methods fail to learn relationships between FJSP nodes, such as interactions between operations of different jobs, leading to limited interpretability and performance. To address these issues, we propose a dual operation aggregation graph neural network (GNN) for solving FJSP. Specifically, we decouple the disjunctive graph into two distinct graphs, reducing graph density and clarifying relationships between machines and operations, thus enabling more effective aggregation and understanding by neural networks. We develop two distinct graph aggregation methods to minimize the influence of non-critical machine and operation nodes on decision-making while enhancing the model's ability to account for long-term benefits. Additionally, to achieve more accurate multi-objective estimation and mitigate reward sparsity, we design a reward function that simultaneously considers machine efficiency, schedule balance, and makespan minimization. Extensive experimental results on well-known datasets demonstrate that our model outperforms state-of-the-art models and exhibits excellent generalization capabilities, effectively addressing the challenges of cloud manufacturing.