Graph neural networks (GNNs) have shown significant success in learning graph representations. However, recent studies reveal that GNNs often fail to outperform simple MLPs on heterophilous graph tasks, where connected nodes may differ in features or labels, challenging the homophily assumption. Existing methods addressing this issue often overlook the importance of information granularity and rarely consider implicit relationships between distant nodes. To overcome these limitations, we propose the Granular and Implicit Graph Network (GRAIN), a novel GNN model specifically designed for heterophilous graphs. GRAIN enhances node embeddings by aggregating multi-view information at various granularity levels and incorporating implicit data from distant, non-neighboring nodes. This approach effectively integrates local and global information, resulting in smoother, more accurate node representations. We also introduce an adaptive graph information aggregator that efficiently combines multi-granularity and implicit data, significantly improving node representation quality, as shown by experiments on 13 datasets covering varying homophily and heterophily. GRAIN consistently outperforms 12 state-of-the-art models, excelling on both homophilous and heterophilous graphs.

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.

Graph Neural Networks (GNNs) have been gaining more attention due to their excellent performance in modeling various graph-structured data. However, most of the current GNNs only consider fixed-neighbor discrete message-passing, disregarding the importance of the local structure of different nodes and the implicit information between nodes for smoothing features. Previous approaches either focus on adaptive selection for aggregation structures or treat discrete graph convolution as a continuous diffusion process, but none of them comprehensively considered the above issues, significantly limiting the model's performance. To this end, we present a novel approach called Flexible Diffusion Convolution (Flexi-DC), which exploits the neighborhood information of nodes to set a particular continuous diffusion for each node to smooth features. Specifically, Flexi-DC first extracts the local structure knowledge based on the degrees of nodes in the graph data and then injects it into the diffusion convolution module to smooth features. Additionally, we utilize the extracted knowledge to smooth labels. Flexi-DC is an efficient framework that can significantly improve the performance of most GNN architectures. Experimental results demonstrate that Flexi-DC outperforms their vanilla implementations by an average accuracy of 13.24% (GCN), 16.37% (JKNet), and 11.98% (ARMA) on nine graph datasets with different homophily ratios.

While performing favourably on the independent and identically distributed (i.i.d.) instances, most of the existing neural methods for vehicle routing problems (VRPs) struggle to generalize in the presence of a distribution shift. To tackle this issue, we propose an ensemble-based deep reinforcement learning method for VRPs, which learns a group of diverse sub-policies to cope with various instance distributions. In particular, to prevent convergence of the parameters to the same one, we enforce diversity across sub-policies by leveraging Bootstrap with random initialization. Moreover, we also explicitly pursue inequality between sub-policies by exploiting regularization terms during training to further enhance diversity. Experimental results show that our method is able to outperform the state-of-the-art neural baselines on randomly generated instances of various distributions, and also generalizes favourably on the benchmark instances from TSPLib and CVRPLib, which confirmed the effectiveness of the whole method and the respective designs.

Recently, neural heuristics based on deep learning have reported encouraging results for solving vehicle routing problems (VRPs), especially on independent and identically distributed (i.i.d.) instances, e.g. uniform. However, in the presence of a distribution shift for the testing instances, their performance becomes considerably inferior. In this paper, we propose a multi-view graph contrastive learning (MVGCL) approach to enhance the generalization across different distributions, which exploits a graph pattern learner in a self-supervised fashion to facilitate a neural heuristic equipped with an active search scheme. Specifically, our MVGCL first leverages graph contrastive learning to extract transferable patterns from VRP graphs to attain the generalizable multi-view (i.e. node and graph) representation. Then it adopts the learnt node embedding and graph embedding to assist the neural heuristic and the active search (during inference) for route construction, respectively. Extensive experiments on randomly generated VRP instances of various distributions, and the ones from TSPLib and CVRPLib show that our MVGCL is superior to the baselines in boosting the cross-distribution generalization performance.

Recently, there is a growing attention on applying deep reinforcement learning (DRL) to solve the 3-D bin packing problem (3-D BPP). However, due to the relatively less informative yet computationally heavy encoder, and considerably large action space inherent to the 3-D BPP, existing DRL methods are only able to handle up to 50 boxes. In this article, we propose to alleviate this issue via a DRL agent, which sequentially addresses three subtasks of sequence, orientation, and position, respectively. Specifically, we exploit a multimodal encoder, where a sparse attention subencoder embeds the box state to mitigate the computation while learning the packing policy, and a convolutional neural network subencoder embeds the view state to produce auxiliary spatial representation. We also leverage an action representation learning in the decoder to cope with the large action space of the position subtask. Besides, we integrate the proposed DRL agent into constraint programming (CP) to further improve the solution quality iteratively by exploiting the powerful search framework in CP. The experiments show that both the sole DRL and hybrid methods enable the agent to solve large-scale instances of 120 boxes or more. Moreover, they both could deliver superior performance against the baselines on instances of various scales.

Recent deep models for solving routing problems always assume a single distribution of nodes for training, which severely impairs their cross-distribution generalization ability. In this paper, we exploit group distributionally robust optimization (group DRO) to tackle this issue, where we jointly optimize the weights for different groups of distributions and the parameters for the deep model in an interleaved manner during training. We also design a module based on convolutional neural network, which allows the deep model to learn more informative latent pattern among the nodes. We evaluate the proposed approach on two types of well-known deep models including GCN and POMO. The experimental results on the randomly synthesized instances and the ones from two benchmark dataset (i.e., TSPLib and CVRPLib) demonstrate that our approach could significantly improve the cross-distribution generalization performance over the original models.