Recent advances in Neural Combinatorial Optimization (NCO) methods have significantly improved the capability of neural solvers to handle synthetic routing instances. Nonetheless, existing neural solvers typically struggle to generalize effectively from synthetic, uniformly-distributed training data to real-world VRP scenarios, including widely recognized benchmark instances from TSPLib and CVRPLib. To bridge this generalization gap, we present Evolutionary Realistic Instance Synthesis (EvoReal), which leverages an evolutionary module guided by large language models (LLMs) to generate synthetic instances characterized by diverse and realistic structural patterns. Specifically, the evolutionary module produces synthetic instances whose structural attributes statistically mimics those observed in authentic real-world instances. Subsequently, pre-trained NCO models are progressively refined, firstly aligning them with these structurally enriched synthetic distributions and then further adapting them through direct fine-tuning on actual benchmark instances. Extensive experimental evaluations demonstrate that EvoReal markedly improves the generalization capabilities of state-of-the-art neural solvers, yielding a notable reduced performance gap compared to the optimal solutions on the TSPLib (1.05%) and CVRPLib (2.71%) benchmarks across a broad spectrum of problem scales.

Multiobjective optimization (MOO) is important for deep learning applications with multiple conflicting objectives. Pareto front learning (PFL) methods learn a single model conditioned on the preference of objectives and can be applied to any preference at inference time. However, existing PFL methods use predefined strategies (e.g., uniform sampling) to generate preferences, which could result in unevenly spaced solutions since the shape of Pareto front is largely ignored. In this article, we propose a lightweight and model-agnostic method to train a preference generator for a given PFL model, which learns to generate proper preferences from uniformly sampled ones, such that the resulting solutions are evenly spaced on the Pareto front. Compared to previous works, our method enables a more rational allocation of preferences, which can either be utilized to enhance a pretrained PFL model or be seamlessly integrated into the PFL training process to improve efficiency. We apply our method to state-of-the-art PFL methods with various backbones (e.g., multilayer perceptron, convolutional neural network, transformer) and validate the significance of preference generation across various tasks, from multitask supervised learning to multiobjective reinforcement learning-based neural combinatorial optimization. Experimental results show that our method improves the backbone algorithm in most settings, showing its effectiveness and general applicability.

Neural combinatorial optimization (NCO) has gained significant attention due to the potential of deep learning to efficiently solve combinatorial optimization problems. NCO has been widely applied to job shop scheduling problems (JSPs) with the current focus predominantly on deterministic problems. In this paper, we propose a novel attention-based scenario processing module (SPM) to extend NCO methods for solving stochastic JSPs. Our approach explicitly incorporates stochastic information by an attention mechanism that captures the embedding of sampled scenarios (i.e., an approximation of stochasticity). Fed with the embedding, the base neural network is intervened by the attended scenarios, which accordingly learns an effective policy under stochasticity. We also propose a training paradigm that works harmoniously with either the expected makespan or Value-at-Risk objective. Results demonstrate that our approach outperforms existing learning and non-learning methods for the flexible JSP problem with stochastic processing times on a variety of instances. In addition, our approach holds significant generalizability to varied numbers of scenarios and disparate distributions.

Deep reinforcement learning (DRL) has emerged as an effective technique for dynamic algorithm configuration, particularly in evolutionary computation, enabling adaptive parameter updates during algorithmic execution. DRL-based methods have shown broad applicability across different problem domains and are designed to configure algorithms without problem-specific information, making them highly transferable across problem variants and scalable to different problem sizes. This paper proposes a novel graph neural network-based approach that learns representations of Search Trajectory Networks (STNs) to track the convergence behavior of multiple objectives and dynamically reconfigures multi-objective evolutionary algorithms during execution. By capturing how solutions evolve and interact over time, the STN-based state representation enables real-time insight into convergence, diversity, and their trade-offs, facilitating more informed and adaptive configuration decisions. Extensive experiments indicate that our method outperforms the state-of-the-art DRL-based algorithm configuration methods. It also demonstrates good scalability to large problem instances and effectiveness in real-world optimization problems, which are often computationally expensive to tune.

Recent decomposition-based neural multi-objective combinatorial optimization (MOCO) methods struggle to achieve desirable performance. Even equipped with complex learning techniques, they often suffer from significant optimality gaps in weight-specific subproblems. To address this challenge, we propose a neat weight embedding method to learn weight-specific representations, which captures weight-instance interaction for the subproblems and was overlooked by most current methods. We demonstrate the potentials of our method in two instantiations. First, we introduce a succinct addition model to learn weight-specific node embeddings, which surpassed most existing neural methods. Second, we design an enhanced conditional attention model to simultaneously learn the weight embedding and node embeddings, which yielded new state-of-the-art performance. Experimental results on classic MOCO problems verified the superiority of our method. Remarkably, our method also exhibits favorable generalization performance across problem sizes, even outperforming the neural method specialized for boosting size generalization.

Existing neural multi-objective combinatorial optimization (MOCO) methods still exhibit an optimality gap since they fail to fully exploit the intrinsic features of problem instances. A significant factor contributing to this shortfall is their reliance solely on graph-modal information. To overcome this, we propose a novel graph-image multimodal fusion (GIMF) framework that enhances neural MOCO methods by integrating graph and image information of the problem instances. Our GIMF framework comprises three key components: (1) a constructed coordinate image to better represent the spatial structure of the problem instance, (2) a problem-size adaptive resolution strategy during the image construction process to improve the cross-size generalization of the model, and (3) a multimodal fusion mechanism with modality-specific bottlenecks to efficiently couple graph and image information. We demonstrate the versatility of our GIMF by implementing it with two state-of-the-art neural MOCO backbones. Experimental results on classic MOCO problems show that our GIMF significantly outperforms state-of-the-art neural MOCO methods and exhibits superior generalization capability.

This paper proposes Decomposed Retrieval of Constraints (DRoC), a novel framework aimed at enhancing large language models (LLMs) in exploiting solvers to tackle vehicle routing problems (VRPs) with intricate constraints. While LLMs have shown promise in solving simple VRPs, their potential in addressing complex VRP variants is still suppressed, due to the limited embedded internal knowledge that is required to accurately reflect diverse VRP constraints. Our approach mitigates the issue by integrating external knowledge via a novel retrieval-augmented generation (RAG) approach. More specifically, the DRoC decomposes VRP constraints, externally retrieves information relevant to each constraint, and synergistically combines internal and external knowledge to benefit the program generation for solving VRPs. The DRoC also allows LLMs to dynamically select between RAG and self-debugging mechanisms, thereby optimizing program generation without the need for additional training. Experiments across 48 VRP variants exhibit the superiority of DRoC, with significant improvements in the accuracy rate and runtime error rate delivered by the generated programs. The DRoC framework has the potential to elevate LLM performance in complex optimization tasks, fostering the applicability of LLMs in industries such as transportation and logistics.

Neural Combinatorial Optimization (NCO) methods have exhibited promising performance in solving Vehicle Routing Problems (VRPs). However, most NCO methods rely on the conventional self-attention mechanism that induces excessive computational complexity, thereby struggling to contend with large-scale VRPs and hindering their practical applicability. In this paper, we propose a lightweight cross-attention mechanism with linear complexity, by which a Transformer network is developed to learn efficient and favorable solutions for large-scale VRPs. We also propose a Self-Improved Training (SIT) algorithm that enables direct model training on large-scale VRP instances, bypassing extensive computational overhead for attaining labels. By iterating solution reconstruction, the Transformer network itself can generate improved partial solutions as pseudo-labels to guide the model training. Experimental results on the Travelling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) with up to 100K nodes indicate that our method consistently achieves superior performance for synthetic and real-world benchmarks, significantly boosting the scalability of NCO methods.

Deep reinforcement learning (DRL) has been widely used for dynamic algorithm configuration, particularly in evolutionary computation, which benefits from the adaptive update of parameters during the algorithmic execution. However, applying DRL to algorithm configuration for multi-objective combinatorial optimization (MOCO) problems remains relatively unexplored. This paper presents a novel graph neural network (GNN) based DRL to configure multi-objective evolutionary algorithms. We model the dynamic algorithm configuration as a Markov decision process, representing the convergence of solutions in the objective space by a graph, with their embeddings learned by a GNN to enhance the state representation. Experiments on diverse MOCO challenges indicate that our method outperforms traditional and DRL-based algorithm configuration methods in terms of efficacy and adaptability. It also exhibits advantageous generalizability across objective types and problem sizes, and applicability to different evolutionary computation methods.

Recent neural heuristics for the Vehicle Routing Problem (VRP) primarily rely on node coordinates as input, which may be less effective in practical scenarios where real cost metrics—such as edge-based distances—are more relevant. To address this limitation, we introduce EFormer, an Edge-based Transformer model that uses edge as the sole input for VRPs. Our approach employs a precoder module with a mixed-score attention mechanism to convert edge information into temporary node embeddings. We also present a parallel encoding strategy characterized by a graph encoder and a node encoder, each responsible for processing graph and node embeddings in distinct feature spaces, respectively. This design yields a more comprehensive representation of the global relationships among edges. In the decoding phase, parallel context embedding and multi-query integration are used to compute separate attention mechanisms over the two encoded embeddings, facilitating efficient path construction. We train EFormer using reinforcement learning in an autoregressive manner. Extensive experiments on the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) reveal that EFormer outperforms established baselines on synthetic datasets, including large-scale and diverse distributions. Moreover, EFormer demonstrates strong generalization on real-world instances from TSPLib and CVRPLib. These findings confirm the effectiveness of EFormer’s core design in solving VRPs.

Recent Deep Reinforcement Learning (DRL) techniques have advanced solutions to Vehicle Routing Problems (VRPs). However, many of these methods focus exclusively on optimizing distance-oriented objectives (i.e., minimizing route length), often overlooking the implicit drivers' preferences for routes. These preferences, which are crucial in practice, are challenging to model using traditional DRL approaches. To address this gap, we propose a preference-based DRL method characterized by its reward design and optimization objective, which is specialized to learn historical route preferences. Our experiments demonstrate that the method aligns generated solutions more closely with human preferences. Moreover, it exhibits strong generalization performance across a variety of instances, offering a robust solution for different VRP scenarios.

Existing neural methods for the Travelling Salesman Problem (TSP) mostly aim at finding a single optimal solution. To discover diverse yet high-quality solutions for Multi-Solution TSP (MSTSP), we propose a novel deep reinforcement learning based neural solver, which is primarily featured by an encoder-decoder structured policy. Concretely, on the one hand, a Relativization Filter (RF) is designed to enhance the robustness of the encoder to affine transformations of the instances, so as to potentially improve the quality of the found solutions. On the other hand, a Multi-Attentive Adaptive Active Search (MA3S) is tailored to allow the decoders to strike a balance between the optimality and diversity. Experimental evaluations on benchmark instances demonstrate the superiority of our method over recent neural baselines across different metrics, and its competitive performance against state-of-the-art traditional heuristics with significantly reduced computational time, ranging from 1.3× to 15× faster. Furthermore, we demonstrate that our method can also be applied to the Capacitated Vehicle Routing Problem (CVRP).

Recent deep reinforcement learning methods have achieved remarkable success in solving multi-objective combinatorial optimization problems (MOCOPs) by decomposing them into multiple subproblems, each associated with a specific weight vector. However, these methods typically treat all subproblems equally and solve them using a single model, hindering the effective exploration of the solution space and thus leading to suboptimal performance. To overcome the limitation, we propose POCCO, a novel plug-and-play framework that enables adaptive selection of model structures for subproblems, which are subsequently optimized based on preference signals rather than explicit reward values. Specifically, we design a conditional computation block that routes subproblems to specialized neural architectures. Moreover, we propose a preference-driven optimization algorithm that learns pairwise preferences between winning and losing solutions. We evaluate the efficacy and versatility of POCCO by applying it to two state-of-the-art neural methods for MOCOPs. Experimental results across four classic MOCOP benchmarks demonstrate its significant superiority and strong generalization.

Combinatorial optimization (CO) problems, central to decision-making scenarios like logistics and manufacturing, are traditionally solved using problem-specific algorithms requiring significant domain expertise. While large language models (LLMs) have shown promise in automating CO problem solving, existing approaches rely on intermediate steps such as code generation or solver invocation, limiting their generality and accessibility. This paper introduces a novel framework that empowers LLMs to serve as end-to-end CO solvers by directly mapping natural language problem descriptions to solutions. We propose a two-stage training strategy: supervised fine-tuning (SFT) imparts LLMs with solution construction patterns from domain-specific solvers, while a feasibility-and-optimality-aware reinforcement learning (FOARL) process explicitly mitigates constraint violations and refines solution quality. Evaluation across seven NP-hard CO problems shows that our method achieves a high feasibility rate and reduces the average optimality gap to 1.03–8.20% by tuning a 7B-parameter LLM, surpassing both general-purpose LLMs (e.g., GPT-4o), reasoning models (e.g., DeepSeek-R1), and domain-specific heuristics. Our method establishes a unified language-based pipeline for CO without extensive code execution or manual architectural adjustments for different problems, offering a general and language-driven alternative to traditional solver design while maintaining relative feasibility guarantees.

Recent neural combinatorial optimization (NCO) methods have shown promising problem-solving ability without requiring domain-specific expertise. Most existing NCO methods use training and testing data with a fixed constraint value and lack research on the effect of constraint tightness on the performance of NCO methods. This paper takes the capacity-constrained vehicle routing problem (CVRP) as an example to empirically analyze the NCO performance under different tightness degrees of the capacity constraint. Our analysis reveals that existing NCO methods overfit the capacity constraint, and they can only perform satisfactorily on a small range of the constraint values but poorly on other values. To tackle this drawback of existing NCO methods, we develop an efficient training scheme that explicitly considers varying degrees of constraint tightness and propose a multi-expert module to learn a generally adaptable solving strategy. Experimental results show that the proposed method can effectively overcome the overfitting issue, demonstrating superior performance on the CVRP and CVRP with time windows (CVRPTW) with various constraint tightness degrees. The code is available at [https://github.com/CIAM-Group/Rethinking\_Constraint\_Tightness](https://github.com/CIAM-Group/Rethinking\_Constraint\_Tightness).

Neural solvers for the Vehicle Routing Problem (VRP) have typically relied on either node or edge inputs, limiting their flexibility and generalization in real-world scenarios. We propose UniteFormer, a unified neural solver that supports node-only, edge-only, and hybrid input types through a single model trained via joint edge-node modalities. UniteFormer introduces: (1) a mixed encoder that integrates graph convolutional networks and attention mechanisms to collaboratively process node and edge features, capturing cross-modal interactions between them; and (2) a parallel decoder enhanced with query mapping and a feed-forward layer for improved representation. The model is trained with REINFORCE by randomly sampling input types across batches. Experiments on the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) demonstrate that UniteFormer achieves state-of-the-art performance and generalizes effectively to TSPLib and CVRPLib instances. These results underscore UniteFormer’s ability to handle diverse input modalities and its strong potential to improve performance across various VRP tasks.

Multi-Task Learning (MTL) in Neural Combinatorial Optimization (NCO) is a promising approach for training a unified model capable of solving multiple Vehicle Routing Problem (VRP) variants. However, existing Reinforcement Learning (RL)-based multi-task methods can only train light decoder models on small-scale problems, exhibiting limited generalization ability when solving large-scale problems. To overcome this limitation, this work introduces a novel multi-task learning method driven by knowledge distillation (MTL-KD), which enables efficient training of heavy decoder models with strong generalization ability. The proposed MTL-KD method transfers policy knowledge from multiple distinct RL-based single-task models to a single heavy decoder model, facilitating label-free training and effectively improving the model's generalization ability across diverse tasks. In addition, we introduce a flexible inference strategy termed Random Reordering Re-Construction (R3C), which is specifically adapted for diverse VRP tasks and further boosts the performance of the multi-task model. Experimental results on 6 seen and 10 unseen VRP variants with up to 1,000 nodes indicate that our proposed method consistently achieves superior performance on both uniform and real-world benchmarks, demonstrating robust generalization abilities. The code is available at [https://github.com/CIAM-Group/MTLKD](https://github.com/CIAM-Group/MTLKD).

Infrared small target detection (IRSTD) based on deep learning has received extensive research and application. However, deep learning models require a large amount of data to perform well, and the collection and standardization of infrared small target data is challenging, limiting the applicability of such models. To address this issue, this study proposes a data augmentation scheme for infrared small targets based on Generative Adversarial Networks (GANs). The proposed method is a two-step approach: the first step is the generation of clean backgrounds, and the second is the adaptive fusion of targets and backgrounds. In the background generation stage, we first use the Fast Marching Method (FMM) to fill background targets and obtain clean backgrounds. Then, we design a multi-generator and multi-discriminator GAN model (MGD-GAN) to generate high-quality and diverse background images. In the adaptive target-background fusion stage, we propose a dual-discriminator GAN network (FusionGAN), which allows the target mask to be adaptively fused with the background pixels. By combining real targets with generated backgrounds, new infrared small target images are generated, achieving the goal of data augmentation. Experiments conducted across three different scenarios demonstrate that the proposed data augmentation scheme effectively enhances the performance of both traditional and advanced detection models.

Existing neural heuristics for multiobjective vehicle routing problems (MOVRPs) are primarily conditioned on instance context, which failed to appropriately exploit preference and problem size, thus holding back the performance. To thoroughly unleash the potential, we propose a novel conditional neural heuristic (CNH) that fully leverages the instance context, preference, and size with an encoder-decoder structured policy network. Particularly, in our CNH, we design a dual-attention-based encoder to relate preferences and instance contexts, so as to better capture their joint effect on approximating the exact Pareto front (PF). We also design a size-aware decoder based on the sinusoidal encoding to explicitly incorporate the problem size into the embedding, so that a single trained model could better solve instances of various scales. Besides, we customize the REINFORCE algorithm to train the neural heuristic by leveraging stochastic preferences (SPs), which further enhances the training performance. Extensive experimental results on random and benchmark instances reveal that our CNH could achieve favorable approximation to the whole PF with higher hypervolume (HV) and lower optimality gap (Gap) than those of the existing neural and conventional heuristics. More importantly, a single trained model of our CNH can outperform other neural heuristics that are exclusively trained on each size. In addition, the effectiveness of the key designs is also verified through ablation studies.

The vehicle routing problem with backhauls (VRPBs) is a challenging problem commonly studied in computer science and operations research. Featured by linehaul (or delivery) and backhaul (or pickup) customers, the VRPB has broad applications in real-world logistics. In this article, we propose a neural heuristic based on deep reinforcement learning (DRL) to solve the traditional and improved VRPB variants, with an encoder-decoder structured policy network trained to sequentially construct the routes for vehicles. Specifically, we first describe the VRPB based on a graph and cast the solution construction as a Markov decision process (MDP). Then, to identify the relationship among the nodes (i.e., linehaul and backhaul customers, and the depot), we design a two-stage attention-based encoder, including a self-attention and a heterogeneous attention for each stage, which could yield more informative representations of the nodes so as to deliver high-quality solutions. The evaluation on the two VRPB variants reveals that, our neural heuristic performs favorably against both the conventional and neural heuristic baselines on randomly generated instances and benchmark instances. Moreover, the trained policy network exhibits a desirable capability of generalization to various problem sizes and distributions.

In evolutionary robotics (ER), the evolution of a robot’s morphology (i.e., physical structure) or controller (i.e., control algorithm or instruction sequence) often entails tackling an extensive number of tasks. The use of evolutionary multitasking (EMT) in ER, which optimizes multiple tasks simultaneously by reusing potentially useful knowledge across diverse tasks, could improve the performance of problem-solving to each task. However, existing EMT methods do not fully use intertask correlations, limiting knowledge sharing. In view of this, this study introduces a novel framework, termed adaptive group-based collaborative optimization, tailored for handling optimization problems involving a large number of tasks within the ER domain simultaneously. The proposed framework divides tasks into groups according to their similarity and then proceeds through two principal stages, namely, intergroup knowledge separation and intragroup knowledge reunion. During intergroup knowledge separation stage, an adaptive method for selecting crossover operators enables source tasks to share useful knowledge to the target task across groups. During intragroup knowledge reunion stage, an adaptive knowledge combination strategy facilitates the target task in assimilating knowledge from multiple sources intragroup. We validated the efficacy of the proposed framework in both planar manipulators and hexapod robot experiments. The results indicate that our method outperforms existing state-of-the-art algorithms (i.e., MME, MMKT) on several metrics (e.g., mean fitness and quality diversity metrics). The proposed method can effectively improve the effectiveness and diversity of solutions in solving ER problems with a large number of tasks (e.g., 5 000 or 10 000), and has broad potential in practical ER applications.

Portfolio selection stands as a paramount concern within the realm of decision-making and management engineering. However, owing to the inherent intricacies of capital markets and the presence of irrational investor behaviors, the attainment of predefined investment objectives by investors remains a formidable challenge. In order to comprehensively depict investor behavior patterns and to provide investment guidance in highly uncertain and volatile markets, this study introduces a novel fuzzy model for representing prospect theory and based on this, develops a novel portfolio selection optimization framework. In addition, a new particle swarm optimization consists of adaptive and cooperative strategy is proposed to find the optimal solution of this model. The effectiveness of this model is validated through two case study utilizing real-market data, while the efficiency of the solution algorithm is confirmed through a test fitness functions-based case study.

Recent studies in using deep reinforcement learning (DRL) to solve Job-shop scheduling problems (JSSP) focus on construction heuristics. However, their performance is still far from optimality, mainly because the underlying graph representation scheme is unsuitable for modelling partial solutions at each construction step. This paper proposes a novel DRL-guided improvement heuristic for solving JSSP, where graph representation is employed to encode complete solutions. We design a Graph-Neural-Network-based representation scheme, consisting of two modules to effectively capture the information of dynamic topology and different types of nodes in graphs encountered during the improvement process. To speed up solution evaluation during improvement, we present a novel message-passing mechanism that can evaluate multiple solutions simultaneously. We prove that the computational complexity of our method scales linearly with problem size. Experiments on classic benchmarks show that the improvement policy learned by our method outperforms state-of-the-art DRL-based methods by a large margin.

Recently, convolutional neural networks have yielded promising results in infrared small target detection. However, limited data is a main restriction to the further promotion of detection performance. To solve this issue, we propose a novel two-stage synthetic data augmentation method, involving StyleGAN-based background generation and Transformer-based target fusion, which aims at generating diverse infrared small target images fitting the original distribution. In the background generation stage, we devise a spatial and low-frequency StyleGAN to ameliorate background generation quality, effectively adapting to less high-frequency information in infrared images. In the target fusion stage, a target prior-based Transformer model with a new detecting difficulty distribution similarity loss is proposed to modulate the intensity of targets implanted on synthetic backgrounds. Experimental results show that our synthetic data augmentation method greatly improves the performance of four detection models on three public datasets and attains state-of-the-art results compared to existing data augmentation methods.

Learning to solve vehicle routing problems (VRPs) has garnered much attention. However, most neural solvers are only structured and trained independently on a specific problem, making them less generic and practical. In this paper, we aim to develop a unified neural solver that can cope with a range of VRP variants simultaneously. Specifically, we propose a multi-task vehicle routing solver with mixture-of-experts (MVMoE), which greatly enhances the model capacity without a proportional increase in computation. We further develop a hierarchical gating mechanism for the MVMoE, delivering a good trade-off between empirical performance and computational complexity. Experimentally, our method significantly promotes zero-shot generalization performance on 10 unseen VRP variants, and showcases decent results on the few-shot setting and real-world benchmark instances. We further conduct extensive studies on the effect of MoE configurations in solving VRPs, and observe the superiority of hierarchical gating when facing out-of-distribution data. The source code is available at: https://github.com/RoyalSkye/Routing-MVMoE.

As a core problem of online games, matchmaking is to assign players into multiple teams to maximize their gaming experience. With the rapid development of game industry, it is increasingly difficulty to explicitly model players' experiences as linear functions. Instead, it is often modeled in a data-driven way by training a neural network. Meanwhile, complex rules must be satisfied to ensure the robustness of matchmaking, which are often described using logical operators. Therefore, matchmaking in practical scenarios is a challenging combinatorial optimization problem with nonlinear objective, linear constraints and logical constraints, which receives much less attention in previous research. In this paper, we propose a novel deep learning method for high-quality matchmaking in real-time. We first cast the problem as standard mixed-integer programming (MIP) by linearizing ReLU networks and logical constraints. Then, based on supervised learning, we design and train a multi-modal graph learning architecture to predict optimal solutions end-to-end from instance data, and solve a surrogate problem to efficiently obtain feasible solutions. Evaluation results on real industry datasets show that our method can deliver near-optimal solutions within 100ms.

Despite enjoying desirable efficiency and reduced reliance on domain expertise, existing neural methods for vehicle routing problems (VRPs) suffer from severe robustness issues — their performance significantly deteriorates on clean instances with crafted perturbations. To enhance robustness, we propose an ensemble-based *Collaborative Neural Framework (CNF)* w.r.t. the defense of neural VRP methods, which is crucial yet underexplored in the literature. Given a neural VRP method, we adversarially train multiple models in a collaborative manner to synergistically promote robustness against attacks, while boosting standard generalization on clean instances. A neural router is designed to adeptly distribute training instances among models, enhancing overall load balancing and collaborative efficacy. Extensive experiments verify the effectiveness and versatility of CNF in defending against various attacks across different neural VRP methods. Notably, our approach also achieves impressive out-of-distribution generalization on benchmark instances.

Vehicle Routing Problems (VRPs) can model many real-world scenarios and often involve complex constraints. While recent neural methods excel in constructing solutions based on feasibility masking, they struggle with handling complex constraints, especially when obtaining the masking itself is NP-hard. In this paper, we propose a novel Proactive Infeasibility Prevention (PIP) framework to advance the capabilities of neural methods towards more complex VRPs. Our PIP integrates the Lagrangian multiplier as a basis to enhance constraint awareness and introduces preventative infeasibility masking to proactively steer the solution construction process. Moreover, we present PIP-D, which employs an auxiliary decoder and two adaptive strategies to learn and predict these tailored masks, potentially enhancing performance while significantly reducing computational costs during training. To verify our PIP designs, we conduct extensive experiments on the highly challenging Traveling Salesman Problem with Time Window (TSPTW), and TSP with Draft Limit (TSPDL) variants under different constraint hardness levels. Notably, our PIP is generic to boost many neural methods, and exhibits both a significant reduction in infeasible rate and a substantial improvement in solution quality.

Existing learning-based methods for solving job shop scheduling problems (JSSP) usually use off-the-shelf GNN models tailored to undirected graphs and neglect the rich and meaningful topological structures of disjunctive graphs (DGs). This paper proposes the topology-aware bidirectional graph attention network (TBGAT), a novel GNN architecture based on the attention mechanism, to embed the DG for solving JSSP in a local search framework. Specifically, TBGAT embeds the DG from a forward and a backward view, respectively, where the messages are propagated by following the different topologies of the views and aggregated via graph attention. Then, we propose a novel operator based on the message-passing mechanism to calculate the forward and backward topological sorts of the DG, which are the features for characterizing the topological structures and exploited by our model. In addition, we theoretically and experimentally show that TBGAT has linear computational complexity to the number of jobs and machines, respectively, strengthening our method’s practical value. Besides, extensive experiments on five synthetic datasets and seven classic benchmarks show that TBGAT achieves new SOTA results by outperforming a wide range of neural methods by a large margin. All the code and data are publicly available online at https://github.com/zcaicaros/TBGAT.

In recent years, there has been a growing trend towards using deep reinforcement learning (DRL) to solve the NP-hard vehicle routing problems (VRPs). While much success has been achieved, most of the previous studies solely focused on single-depot VRPs, which became less effective in handling more practical scenarios, such as multi-depot VRPs. Although there are many preprocessing measures, such as natural decomposition, those scenarios are still more challenging to optimize. To resolve this issue, we propose the multi-depot multi-type attention (MD-MTA) to solve the multi-depot VRP (MDVRP) and multi-depot open VRP (MDOVRP), respectively. We design a multi-type attention in the network to combine different types of embeddings and the state of the environment at each step, so as to accurately select the next node to visit and construct the route. We introduce a depot rotation augmentation to enhance solution decoding. Results show that it performs favorably against various representative traditional baselines and DRL-based baselines.

While the encoder-decoder structure is widely used in the recent neural construction methods for learning to solve vehicle routing problems (VRPs), they are less effective in searching solutions due to deterministic feature embeddings and deterministic probability distributions. In this article, we propose the feature embedding refiner (FER) with a novel and generic encoder-refiner-decoder structure to boost the existing encoder-decoder structured deep models. It is model-agnostic that the encoder and the decoder can be from any pretrained neural construction method. Regarding the introduced refiner network, we design its architecture by combining the standard gated recurrent units (GRU) cell with two new layers, i.e., an accumulated graph attention (AGA) layer and a gated nonlinear (GNL) layer. The former extracts dynamic graph topological information of historical solutions stored in a diversified solution pool to generate aggregated pool embeddings that are further improved by the GRU, and the latter adaptively refines the feature embeddings from the encoder with the guidance of the improved pool embeddings. To this end, our FER allows current neural construction methods to not only iteratively refine the feature embeddings for boarder search range but also dynamically update the probability distributions for more diverse search. We apply FER to two prevailing neural construction methods including attention model (AM) and policy optimization with multiple optima (POMO) to solve the traveling salesman problem (TSP) and the capacitated VRP (CVRP). Experimental results show that our method achieves lower gaps and better generalization than the original ones and also exhibits competitive performance to the state-of-the-art neural improvement methods.

Learning heuristics for vehicle routing problems (VRPs) has gained much attention due to the less reliance on hand-crafted rules. However, existing methods are typically trained and tested on the same task with a fixed size and distribution (of nodes), and hence suffer from limited generalization performance. This paper studies a challenging yet realistic setting, which considers generalization across both size and distribution in VRPs. We propose a generic meta-learning framework, which enables effective training of an initialized model with the capability of fast adaptation to new tasks during inference. We further develop a simple yet efficient approximation method to reduce the training overhead. Extensive experiments on both synthetic and benchmark instances of the traveling salesman problem (TSP) and capacitated vehicle routing problem (CVRP) demonstrate the effectiveness of our method. The code is available at: https://github.com/RoyalSkye/Omni-VRP.

Most of existing neural methods for multi-objective combinatorial optimization (MOCO) problems solely rely on decomposition, which often leads to repetitive solutions for the respective subproblems, thus a limited Pareto set. Beyond decomposition, we propose a novel neural heuristic with diversity enhancement (NHDE) to produce more Pareto solutions from two perspectives. On the one hand, to hinder duplicated solutions for different subproblems, we propose an indicator-enhanced deep reinforcement learning method to guide the model, and design a heterogeneous graph attention mechanism to capture the relations between the instance graph and the Pareto front graph. On the other hand, to excavate more solutions in the neighborhood of each subproblem, we present a multiple Pareto optima strategy to sample and preserve desirable solutions. Experimental results on classic MOCO problems show that our NHDE is able to generate a Pareto front with higher diversity, thereby achieving superior overall performance. Moreover, our NHDE is generic and can be applied to different neural methods for MOCO.

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.

Instance-specific Algorithm Configuration (AC) methods are effective in automatically generating high-quality algorithm parameters for heterogeneous NP-hard problems from multiple sources. However, existing works rely on manually designed features to describe training instances, which are simple numerical attributes and cannot fully capture structural differences. Targeting at Mixed-Integer Programming (MIP) solvers, this paper proposes a novel instances-specific AC method based on end-to-end deep graph clustering. By representing an MIP instance as a bipartite graph, a random walk algorithm is designed to extract raw features with both numerical and structural information from the instance graph. Then an auto-encoder is designed to learn dense instance embeddings unsupervisedly, which facilitates clustering heterogeneous instances into homogeneous clusters for training instance-specific configurations. Experimental results on multiple benchmarks show that the proposed method can improve the solving efficiency of CPLEX on highly heterogeneous instances, and outperform existing instance specific AC methods.

Airport ground handling (AGH) offers necessary operations to flights during their turnarounds and is of great importance to the efficiency of airport management and the economics of aviation. Such a problem involves the interplay among the operations that leads to NP-hard problems with complex constraints. Hence, existing methods for AGH are usually designed with massive domain knowledge but still fail to yield high-quality solutions efficiently. In this paper, we aim to enhance the solution quality and computation efficiency for solving AGH. Particularly, we first model AGH as a multiple-fleet vehicle routing problem (VRP) with miscellaneous constraints including precedence, time windows, and capacity. Then we propose a construction framework that decomposes AGH into sub-problems (i.e., VRPs) in fleets and present a neural method to construct the routing solutions to these sub-problems. In specific, we resort to deep learning and parameterize the construction heuristic policy with an attention-based neural network trained with reinforcement learning, which is shared across all sub-problems. Extensive experiments demonstrate that our method significantly outperforms classic meta-heuristics, construction heuristics and the specialized methods for AGH. Besides, we empirically verify that our neural method generalizes well to instances with large numbers of flights or varying parameters, and can be readily adapted to solve real-time AGH with stochastic flight arrivals. Our code is publicly available at:https://github.com/RoyalSkye/AGH.

Dispatching vehicle fleets to serve flights is a key task in airport ground handling (AGH). Due to the notable growth of flights, it is challenging to simultaneously schedule multiple types of operations (services) for a large number of flights, where each type of operation is performed by one specific vehicle fleet. To tackle this issue, we first represent the operation scheduling as a complex vehicle routing problem and formulate it as a mixed integer linear programming (MILP) model. Then given the graph representation of the MILP model, we propose a learning assisted large neighborhood search (LNS) method using data generated based on real scenarios, where we integrate imitation learning and graph convolutional network (GCN) to learn a destroy operator to automatically select variables, and employ an off-the-shelf solver as the repair operator to reoptimize the selected variables. Experimental results based on a real airport show that the proposed method allows for handling up to 200 flights with 10 types of operations simultaneously, and outperforms state-of-the-art methods. Moreover, the learned method performs consistently accompanying different solvers, and generalizes well on larger instances, verifying the versatility and scalability of our method.

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.

Objective-space decomposition algorithms (ODAs) are widely studied for solving multi-objective integer programs. However, they often encounter difficulties in handling scalarized problems, which could cause infeasibility or repetitive nondominated points and thus induce redundant runtime. To mitigate the issue, we present a graph neural network (GNN) based method to learn the reduction rule in the ODA. We formulate the algorithmic procedure of generic ODAs as a Markov decision process, and parameterize the policy (reduction rule) with a novel two-stage GNN to fuse information from variables, constraints and especially objectives for better state representation. We train our model with imitation learning and deploy it on a state-of-the-art ODA. Results show that our method significantly improves the solving efficiency of the ODA. The learned policy generalizes fairly well to larger problems or more objectives, and the proposed GNN outperforms existing ones for integer programming in terms of test and generalization accuracy.

Recent studies in using deep learning (DL) to solve routing problems focus on construction heuristics, whose solutions are still far from optimality. Improvement heuristics have great potential to narrow this gap by iteratively refining a solution. However, classic improvement heuristics are all guided by handcrafted rules that may limit their performance. In this article, we propose a deep reinforcement learning framework to learn the improvement heuristics for routing problems. We design a self-attention-based deep architecture as the policy network to guide the selection of the next solution. We apply our method to two important routing problems, i.e., the traveling salesman problem (TSP) and the capacitated vehicle routing problem (CVRP). Experiments show that our method outperforms state-of-the-art DL-based approaches. The learned policies are more effective than the traditional handcrafted ones and can be further enhanced by simple diversifying strategies. Moreover, the policies generalize well to different problem sizes, initial solutions, and even real-world data set.

We propose a deep reinforcement learning (RL) method to learn large neighborhood search (LNS) policy for integer programming (IP). The RL policy is trained as the destroy operator to select a subset of variables at each step, which is reoptimized by an IP solver as the repair operator. However, the combinatorial number of variable subsets prevents direct application of typical RL algorithms. To tackle this challenge, we represent all subsets by factorizing them into binary decisions on each variable. We then design a neural network to learn policies for each variable in parallel, trained by a customized actor-critic algorithm. We evaluate the proposed method on four representative IP problems. Results show that it can find better solutions than SCIP in much less time, and significantly outperform other LNS baselines with the same runtime. Moreover, these advantages notably persist when the policies generalize to larger problems. Further experiments with Gurobi also reveal that our method can outperform this state-of-the-art commercial solver within the same time limit.