Zhiguang Cao

Yaoxin Wu, Jianan Zhou, Yunwen Xia et al.

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.

Yining Ma, Zhiguang Cao, Yeow Meng Chee

In this paper, we present Neural k-Opt (NeuOpt), a novel learning-to-search (L2S) solver for routing problems. It learns to perform flexible k-opt exchanges based on a tailored action factorization method and a customized recurrent dual-stream decoder. As a pioneering work to circumvent the pure feasibility masking scheme and enable the autonomous exploration of both feasible and infeasible regions, we then propose the Guided Infeasible Region Exploration (GIRE) scheme, which supplements the NeuOpt policy network with feasibility-related features and leverages reward shaping to steer reinforcement learning more effectively. Additionally, we equip NeuOpt with Dynamic Data Augmentation (D2A) for more diverse searches during inference. Extensive experiments on the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) demonstrate that our NeuOpt not only significantly outstrips existing (masking-based) L2S solvers, but also showcases superiority over the learning-to-construct (L2C) and learning-to-predict (L2P) solvers. Notably, we offer fresh perspectives on how neural solvers can handle VRP constraints. Our code is available: https://github.com/yining043/NeuOpt.

Haoran Ye, Jiarui Wang, Zhiguang Cao et al. · pku

Ant Colony Optimization (ACO) is a meta-heuristic algorithm that has been successfully applied to various Combinatorial Optimization Problems (COPs). Traditionally, customizing ACO for a specific problem requires the expert design of knowledge-driven heuristics. In this paper, we propose DeepACO, a generic framework that leverages deep reinforcement learning to automate heuristic designs. DeepACO serves to strengthen the heuristic measures of existing ACO algorithms and dispense with laborious manual design in future ACO applications. As a neural-enhanced meta-heuristic, DeepACO consistently outperforms its ACO counterparts on eight COPs using a single neural architecture and a single set of hyperparameters. As a Neural Combinatorial Optimization method, DeepACO performs better than or on par with problem-specific methods on canonical routing problems. Our code is publicly available at https://github.com/henry-yeh/DeepACO.

Zeyuan Ma, Hongshu Guo, Jiacheng Chen et al.

Recently, Meta-Black-Box Optimization with Reinforcement Learning (MetaBBO-RL) has showcased the power of leveraging RL at the meta-level to mitigate manual fine-tuning of low-level black-box optimizers. However, this field is hindered by the lack of a unified benchmark. To fill this gap, we introduce MetaBox, the first benchmark platform expressly tailored for developing and evaluating MetaBBO-RL methods. MetaBox offers a flexible algorithmic template that allows users to effortlessly implement their unique designs within the platform. Moreover, it provides a broad spectrum of over 300 problem instances, collected from synthetic to realistic scenarios, and an extensive library of 19 baseline methods, including both traditional black-box optimizers and recent MetaBBO-RL methods. Besides, MetaBox introduces three standardized performance metrics, enabling a more thorough assessment of the methods. In a bid to illustrate the utility of MetaBox for facilitating rigorous evaluation and in-depth analysis, we carry out a wide-ranging benchmarking study on existing MetaBBO-RL methods. Our MetaBox is open-source and accessible at: https://github.com/GMC-DRL/MetaBox.

Rongkai Zhang, Cong Zhang, Zhiguang Cao et al.

We propose a manager-worker framework based on deep reinforcement learning to tackle a hard yet nontrivial variant of Travelling Salesman Problem (TSP), \ie~multiple-vehicle TSP with time window and rejections (mTSPTWR), where customers who cannot be served before the deadline are subject to rejections. Particularly, in the proposed framework, a manager agent learns to divide mTSPTWR into sub-routing tasks by assigning customers to each vehicle via a Graph Isomorphism Network (GIN) based policy network. A worker agent learns to solve sub-routing tasks by minimizing the cost in terms of both tour length and rejection rate for each vehicle, the maximum of which is then fed back to the manager agent to learn better assignments. Experimental results demonstrate that the proposed framework outperforms strong baselines in terms of higher solution quality and shorter computation time. More importantly, the trained agents also achieve competitive performance for solving unseen larger instances.

Jianan Zhou, Yaoxin Wu, Zhiguang Cao et al.

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.

Cong Zhang, Zhiguang Cao, Wen Song et al.

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.

Jieyi Bi, Yining Ma, Jiahai Wang et al.

Recent neural methods for vehicle routing problems always train and test the deep models on the same instance distribution (i.e., uniform). To tackle the consequent cross-distribution generalization concerns, we bring the knowledge distillation to this field and propose an Adaptive Multi-Distribution Knowledge Distillation (AMDKD) scheme for learning more generalizable deep models. Particularly, our AMDKD leverages various knowledge from multiple teachers trained on exemplar distributions to yield a light-weight yet generalist student model. Meanwhile, we equip AMDKD with an adaptive strategy that allows the student to concentrate on difficult distributions, so as to absorb hard-to-master knowledge more effectively. Extensive experimental results show that, compared with the baseline neural methods, our AMDKD is able to achieve competitive results on both unseen in-distribution and out-of-distribution instances, which are either randomly synthesized or adopted from benchmark datasets (i.e., TSPLIB and CVRPLIB). Notably, our AMDKD is generic, and consumes less computational resources for inference.

Yining Ma, Jingwen Li, Zhiguang Cao et al.

We present an efficient Neural Neighborhood Search (N2S) approach for pickup and delivery problems (PDPs). In specific, we design a powerful Synthesis Attention that allows the vanilla self-attention to synthesize various types of features regarding a route solution. We also exploit two customized decoders that automatically learn to perform removal and reinsertion of a pickup-delivery node pair to tackle the precedence constraint. Additionally, a diversity enhancement scheme is leveraged to further ameliorate the performance. Our N2S is generic, and extensive experiments on two canonical PDP variants show that it can produce state-of-the-art results among existing neural methods. Moreover, it even outstrips the well-known LKH3 solver on the more constrained PDP variant. Our implementation for N2S is available online.

Jinbiao Chen, Zizhen Zhang, Zhiguang Cao et al.

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.

Jinbiao Chen, Jiahai Wang, Zizhen Zhang et al.

Recently, neural heuristics based on deep reinforcement learning have exhibited promise in solving multi-objective combinatorial optimization problems (MOCOPs). However, they are still struggling to achieve high learning efficiency and solution quality. To tackle this issue, we propose an efficient meta neural heuristic (EMNH), in which a meta-model is first trained and then fine-tuned with a few steps to solve corresponding single-objective subproblems. Specifically, for the training process, a (partial) architecture-shared multi-task model is leveraged to achieve parallel learning for the meta-model, so as to speed up the training; meanwhile, a scaled symmetric sampling method with respect to the weight vectors is designed to stabilize the training. For the fine-tuning process, an efficient hierarchical method is proposed to systematically tackle all the subproblems. Experimental results on the multi-objective traveling salesman problem (MOTSP), multi-objective capacitated vehicle routing problem (MOCVRP), and multi-objective knapsack problem (MOKP) show that, EMNH is able to outperform the state-of-the-art neural heuristics in terms of solution quality and learning efficiency, and yield competitive solutions to the strong traditional heuristics while consuming much shorter time.

Yong Liang Goh, Zhiguang Cao, Yining Ma et al.

Existing neural constructive solvers for routing problems have predominantly employed transformer architectures, conceptualizing the route construction as a set-to-sequence learning task. However, their efficacy has primarily been demonstrated on entirely random problem instances that inadequately capture real-world scenarios. In this paper, we introduce realistic Traveling Salesman Problem (TSP) scenarios relevant to industrial settings and derive the following insights: (1) The optimal next node (or city) to visit often lies within proximity to the current node, suggesting the potential benefits of biasing choices based on current locations. (2) Effectively solving the TSP requires robust tracking of unvisited nodes and warrants succinct grouping strategies. Building upon these insights, we propose integrating a learnable choice layer inspired by Hypernetworks to prioritize choices based on the current location, and a learnable approximate clustering algorithm inspired by the Expectation-Maximization algorithm to facilitate grouping the unvisited cities. Together, these two contributions form a hierarchical approach towards solving the realistic TSP by considering both immediate local neighbourhoods and learning an intermediate set of node representations. Our hierarchical approach yields superior performance compared to both classical and recent transformer models, showcasing the efficacy of the key designs.

Jianan Zhou, Zhiguang Cao, Yaoxin Wu et al.

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.

Ziwei Huang, Jianan Zhou, Zhiguang Cao et al.

Light decoder-based solvers have gained popularity for solving vehicle routing problems (VRPs) due to their efficiency and ease of integration with reinforcement learning algorithms. However, they often struggle with generalization to larger problem instances or different VRP variants. This paper revisits light decoder-based approaches, analyzing the implications of their reliance on static embeddings and the inherent challenges that arise. Specifically, we demonstrate that in the light decoder paradigm, the encoder is implicitly tasked with capturing information for all potential decision scenarios during solution construction within a single set of embeddings, resulting in high information density. Furthermore, our empirical analysis reveals that the overly simplistic decoder struggles to effectively utilize this dense information, particularly as task complexity increases, which limits generalization to out-of-distribution (OOD) settings. Building on these insights, we show that enhancing the decoder capacity, with a simple addition of identity mapping and a feed-forward layer, can considerably alleviate the generalization issue. Experimentally, our method significantly enhances the OOD generalization of light decoder-based approaches on large-scale instances and complex VRP variants, narrowing the gap with the heavy decoder paradigm. Our code is available at: https://github.com/ziweileonhuang/reld-nco.

Jianghan Zhu, Yaoxin Wu, Zhuoyi Lin et al.

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.

Mingzhao Wang, You Zhou, Zhiguang Cao et al.

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. Our code is available at $\href{https://github.com/DEITSP/DEITSP}{https://github.com/DEITSP/DEITSP}$.

Zeyuan Ma, Zhiguang Cao, Zhou Jiang et al.

Recent progress in Meta-Black-Box-Optimization (MetaBBO) has demonstrated that using RL to learn a meta-level policy for dynamic algorithm configuration (DAC) over an optimization task distribution could significantly enhance the performance of the low-level BBO algorithm. However, the online learning paradigms in existing works makes the efficiency of MetaBBO problematic. To address this, we propose an offline learning-based MetaBBO framework in this paper, termed Q-Mamba, to attain both effectiveness and efficiency in MetaBBO. Specifically, we first transform DAC task into long-sequence decision process. This allows us further introduce an effective Q-function decomposition mechanism to reduce the learning difficulty within the intricate algorithm configuration space. Under this setting, we propose three novel designs to meta-learn DAC policy from offline data: we first propose a novel collection strategy for constructing offline DAC experiences dataset with balanced exploration and exploitation. We then establish a decomposition-based Q-loss that incorporates conservative Q-learning to promote stable offline learning from the offline dataset. To further improve the offline learning efficiency, we equip our work with a Mamba architecture which helps long-sequence learning effectiveness and efficiency by selective state model and hardware-aware parallel scan respectively. Through extensive benchmarking, we observe that Q-Mamba achieves competitive or even superior performance to prior online/offline baselines, while significantly improving the training efficiency of existing online baselines. We provide sourcecodes of Q-Mamba at https://github.com/MetaEvo/Q-Mamba.

Cong Zhang, Zhiguang Cao, Yaoxin Wu et al.

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.

Xian-Rong Zhang, Yue-Jiao Gong, Zhiguang Cao et al.

In recent years, there has been a growing interest in data-driven evolutionary algorithms (DDEAs) employing surrogate models to approximate the objective functions with limited data. However, current DDEAs are primarily designed for lower-dimensional problems and their performance drops significantly when applied to large-scale optimization problems (LSOPs). To address the challenge, this paper proposes an offline DDEA named DSKT-DDEA. DSKT-DDEA leverages multiple islands that utilize different data to establish diverse surrogate models, fostering diverse subpopulations and mitigating the risk of premature convergence. In the intra-island optimization phase, a semi-supervised learning method is devised to fine-tune the surrogates. It not only facilitates data argumentation, but also incorporates the distribution information gathered during the search process to align the surrogates with the evolving local landscapes. Then, in the inter-island knowledge transfer phase, the algorithm incorporates an adaptive strategy that periodically transfers individual information and evaluates the transfer effectiveness in the new environment, facilitating global optimization efficacy. Experimental results demonstrate that our algorithm is competitive with state-of-the-art DDEAs on problems with up to 1000 dimensions, while also exhibiting decent parallelism and scalability. Our DSKT-DDEA is open-source and accessible at: https://github.com/LabGong/DSKT-DDEA.

Haoran Ye, Jiarui Wang, Zhiguang Cao et al. · pku

The omnipresence of NP-hard combinatorial optimization problems (COPs) compels domain experts to engage in trial-and-error heuristic design. The long-standing endeavor of design automation has gained new momentum with the rise of large language models (LLMs). This paper introduces Language Hyper-Heuristics (LHHs), an emerging variant of Hyper-Heuristics that leverages LLMs for heuristic generation, featuring minimal manual intervention and open-ended heuristic spaces. To empower LHHs, we present Reflective Evolution (ReEvo), a novel integration of evolutionary search for efficiently exploring the heuristic space, and LLM reflections to provide verbal gradients within the space. Across five heterogeneous algorithmic types, six different COPs, and both white-box and black-box views of COPs, ReEvo yields state-of-the-art and competitive meta-heuristics, evolutionary algorithms, heuristics, and neural solvers, while being more sample-efficient than prior LHHs.

Zeyuan Ma, Wenqi Huang, Guo-Huan Song et al.

Machine intelligence marks the ultimate dream of making machines' intelligence comparable to human beings. While recent progress in Large Language Models (LLMs) show substantial specific skills for a wide array of downstream tasks, they more or less fall shorts in general intelligence. Following correlation between intelligence and system 2 reasoning (slow thinking), in this paper, we aim to answering a worthwhile research question: could machine intelligence such as LLMs be evolved to acquire reasoning ability (not specific skill) just like our human beings? To this end, we propose evolutionary reasoning optimization (ERO) framework which performs survival of the fittest over a population of LLMs to search for individual with strong reasoning ability. Given a reasoning task, ERO first initializes multiple LLMs as a population, after which an evolutionary strategy evolves the population to maximize quantified reasoning score of the best individual. Based on experiments on representative testsuites, we claim two surprising empirical discoveries: i) the latest LLMs such as GPT-5 still show limited system 2 reasoning ability; ii) with simple evolution-loop of ERO, a relatively weak model (Qwen-7B) could be enhanced to emerge powerful reasoning ability. Our project can be accessed at https://github.com/MetaEvo/ERO for reproduction needs.

Jianan Zhou, Yaoxin Wu, Wen Song et al.

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.

Haoran Ye, Jiarui Wang, Helan Liang et al. · pku

The recent end-to-end neural solvers have shown promise for small-scale routing problems but suffered from limited real-time scaling-up performance. This paper proposes GLOP (Global and Local Optimization Policies), a unified hierarchical framework that efficiently scales toward large-scale routing problems. GLOP partitions large routing problems into Travelling Salesman Problems (TSPs) and TSPs into Shortest Hamiltonian Path Problems. For the first time, we hybridize non-autoregressive neural heuristics for coarse-grained problem partitions and autoregressive neural heuristics for fine-grained route constructions, leveraging the scalability of the former and the meticulousness of the latter. Experimental results show that GLOP achieves competitive and state-of-the-art real-time performance on large-scale routing problems, including TSP, ATSP, CVRP, and PCTSP.

Zeyuan Ma, Hongshu Guo, Jiacheng Chen et al.

Recent research explores optimization using large language models (LLMs) by either iteratively seeking next-step solutions from LLMs or directly prompting LLMs for an optimizer. However, these approaches exhibit inherent limitations, including low operational efficiency, high sensitivity to prompt design, and a lack of domain-specific knowledge. We introduce LLaMoCo, the first instruction-tuning framework designed to adapt LLMs for solving optimization problems in a code-to-code manner. Specifically, we establish a comprehensive instruction set containing well-described problem prompts and effective optimization codes. We then develop a novel two-phase learning strategy that incorporates a contrastive learning-based warm-up procedure before the instruction-tuning phase to enhance the convergence behavior during model fine-tuning. The experiment results demonstrate that a CodeGen (350M) model fine-tuned by our LLaMoCo achieves superior optimization performance compared to GPT-4 Turbo and the other competitors across both synthetic and realistic problem sets. The fine-tuned model and the usage instructions are available at https://anonymous.4open.science/r/LLaMoCo-722A.

Hongshu Guo, Yining Ma, Zeyuan Ma et al.

Evolutionary algorithms, such as Differential Evolution, excel in solving real-parameter optimization challenges. However, the effectiveness of a single algorithm varies across different problem instances, necessitating considerable efforts in algorithm selection or configuration. This paper aims to address the limitation by leveraging the complementary strengths of a group of algorithms and dynamically scheduling them throughout the optimization progress for specific problems. We propose a deep reinforcement learning-based dynamic algorithm selection framework to accomplish this task. Our approach models the dynamic algorithm selection a Markov Decision Process, training an agent in a policy gradient manner to select the most suitable algorithm according to the features observed during the optimization process. To empower the agent with the necessary information, our framework incorporates a thoughtful design of landscape and algorithmic features. Meanwhile, we employ a sophisticated deep neural network model to infer the optimal action, ensuring informed algorithm selections. Additionally, an algorithm context restoration mechanism is embedded to facilitate smooth switching among different algorithms. These mechanisms together enable our framework to seamlessly select and switch algorithms in a dynamic online fashion. Notably, the proposed framework is simple and generic, offering potential improvements across a broad spectrum of evolutionary algorithms. As a proof-of-principle study, we apply this framework to a group of Differential Evolution algorithms. The experimental results showcase the remarkable effectiveness of the proposed framework, not only enhancing the overall optimization performance but also demonstrating favorable generalization ability across different problem classes.

Jieyi Bi, Yining Ma, Jianan Zhou et al.

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.

Zhuoyi Lin, Yaoxin Wu, Bangjian Zhou et al.

Existing neural heuristics often train a deep architecture from scratch for each specific vehicle routing problem (VRP), ignoring the transferable knowledge across different VRP variants. This paper proposes the cross-problem learning to assist heuristics training for different downstream VRP variants. Particularly, we modularize neural architectures for complex VRPs into 1) the backbone Transformer for tackling the travelling salesman problem (TSP), and 2) the additional lightweight modules for processing problem-specific features in complex VRPs. Accordingly, we propose to pre-train the backbone Transformer for TSP, and then apply it in the process of fine-tuning the Transformer models for each target VRP variant. On the one hand, we fully fine-tune the trained backbone Transformer and problem-specific modules simultaneously. On the other hand, we only fine-tune small adapter networks along with the modules, keeping the backbone Transformer still. Extensive experiments on typical VRPs substantiate that 1) the full fine-tuning achieves significantly better performance than the one trained from scratch, and 2) the adapter-based fine-tuning also delivers comparable performance while being notably parameter-efficient. Furthermore, we empirically demonstrate the favorable effect of our method in terms of cross-distribution application and versatility.

Hongshu Guo, Zeyuan Ma, Jiacheng Chen et al.

Recent advances in Meta-learning for Black-Box Optimization (MetaBBO) have shown the potential of using neural networks to dynamically configure evolutionary algorithms (EAs), enhancing their performance and adaptability across various BBO instances. However, they are often tailored to a specific EA, which limits their generalizability and necessitates retraining or redesigns for different EAs and optimization problems. To address this limitation, we introduce ConfigX, a new paradigm of the MetaBBO framework that is capable of learning a universal configuration agent (model) for boosting diverse EAs. To achieve so, our ConfigX first leverages a novel modularization system that enables the flexible combination of various optimization sub-modules to generate diverse EAs during training. Additionally, we propose a Transformer-based neural network to meta-learn a universal configuration policy through multitask reinforcement learning across a designed joint optimization task space. Extensive experiments verify that, our ConfigX, after large-scale pre-training, achieves robust zero-shot generalization to unseen tasks and outperforms state-of-the-art baselines. Moreover, ConfigX exhibits strong lifelong learning capabilities, allowing efficient adaptation to new tasks through fine-tuning. Our proposed ConfigX represents a significant step toward an automatic, all-purpose configuration agent for EAs.

Shipei Zhou, Yuandong Ding, Chi Zhang et al.

This paper proposes a dual divide-and-optimize algorithm (DualOpt) for solving the large-scale traveling salesman problem (TSP). DualOpt combines two complementary strategies to improve both solution quality and computational efficiency. The first strategy is a grid-based divide-and-conquer procedure that partitions the TSP into smaller sub-problems, solving them in parallel and iteratively refining the solution by merging nodes and partial routes. The process continues until only one grid remains, yielding a high-quality initial solution. The second strategy involves a path-based divide-and-optimize procedure that further optimizes the solution by dividing it into sub-paths, optimizing each using a neural solver, and merging them back to progressively improve the overall solution. Extensive experiments conducted on two groups of TSP benchmark instances, including randomly generated instances with up to 100,000 nodes and real-world datasets from TSPLIB, demonstrate the effectiveness of DualOpt. The proposed DualOpt achieves highly competitive results compared to 10 state-of-the-art algorithms in the literature. In particular, DualOpt achieves an improvement gap up to 1.40% for the largest instance TSP100K with a remarkable 104x speed-up over the leading heuristic solver LKH3. Additionally, DualOpt demonstrates strong generalization on TSPLIB benchmarks, confirming its capability to tackle diverse real-world TSP applications.

Chen Wang, Yue-Jiao Gong, Zhiguang Cao et al.

To relieve intensive human-expertise required to design optimization algorithms, recent Meta-Black-Box Optimization (MetaBBO) researches leverage generalization strength of meta-learning to train neural network-based algorithm design policies over a predefined training problem set, which automates the adaptability of the low-level optimizers on unseen problem instances. Currently, a common training problem set choice in existing MetaBBOs is well-known benchmark suites CoCo-BBOB. Although such choice facilitates the MetaBBO's development, problem instances in CoCo-BBOB are more or less limited in diversity, raising the risk of overfitting of MetaBBOs, which might further results in poor generalization. In this paper, we propose an instance generation approach, termed as \textbf{LSRE}, which could generate diverse training problem instances for MetaBBOs to learn more generalizable policies. LSRE first trains an autoencoder which maps high-dimensional problem features into a 2-dimensional latent space. Uniform-grid sampling in this latent space leads to hidden representations of problem instances with sufficient diversity. By leveraging a genetic-programming approach to search function formulas with minimal L2-distance to these hidden representations, LSRE reverse engineers a diversified problem set, termed as \textbf{Diverse-BBO}. We validate the effectiveness of LSRE by training various MetaBBOs on Diverse-BBO and observe their generalization performances on either synthetic or realistic scenarios. Extensive experimental results underscore the superiority of Diverse-BBO to existing training set choices in MetaBBOs. Further ablation studies not only demonstrate the effectiveness of design choices in LSRE, but also reveal interesting insights on instance diversity and MetaBBO's generalization.

Yong Liang Goh, Zhiguang Cao, Yining Ma et al.

Recent advances toward foundation models for routing problems have shown great potential of a unified deep model for various VRP variants. However, they overlook the complex real-world customer distributions. In this work, we advance the Multi-Task VRP (MTVRP) setting to the more realistic yet challenging Multi-Task Multi-Distribution VRP (MTMDVRP) setting, and introduce SHIELD, a novel model that leverages both sparsity and hierarchy principles. Building on a deeper decoder architecture, we first incorporate the Mixture-of-Depths (MoD) technique to enforce sparsity. This improves both efficiency and generalization by allowing the model to dynamically select nodes to use or skip each decoder layer, providing the needed capacity to adaptively allocate computation for learning the task/distribution specific and shared representations. We also develop a context-based clustering layer that exploits the presence of hierarchical structures in the problems to produce better local representations. These two designs inductively bias the network to identify key features that are common across tasks and distributions, leading to significantly improved generalization on unseen ones. Our empirical results demonstrate the superiority of our approach over existing methods on 9 real-world maps with 16 VRP variants each.

Yuxin Pan, Ruohong Liu, Yize Chen et al.

Neural solvers based on the divide-and-conquer approach for Vehicle Routing Problems (VRPs) in general, and capacitated VRP (CVRP) in particular, integrates the global partition of an instance with local constructions for each subproblem to enhance generalization. However, during the global partition phase, misclusterings within subgraphs have a tendency to progressively compound throughout the multi-step decoding process of the learning-based partition policy. This suboptimal behavior in the global partition phase, in turn, may lead to a dramatic deterioration in the performance of the overall decomposition-based system, despite using optimal local constructions. To address these challenges, we propose a versatile Hierarchical Learning-based Graph Partition (HLGP) framework, which is tailored to benefit the partition of CVRP instances by synergistically integrating global and local partition policies. Specifically, the global partition policy is tasked with creating the coarse multi-way partition to generate the sequence of simpler two-way partition subtasks. These subtasks mark the initiation of the subsequent K local partition levels. At each local partition level, subtasks exclusive for this level are assigned to the local partition policy which benefits from the insensitive local topological features to incrementally alleviate the compounded errors. This framework is versatile in the sense that it optimizes the involved partition policies towards a unified objective harmoniously compatible with both reinforcement learning (RL) and supervised learning (SL). (*Due to the notification of arXiv "The Abstract field cannot be longer than 1,920 characters", the appeared Abstract is shortened. For the full Abstract, please download the Article.)

Yunzhuang Shen, Yuan Sun, Xiaodong Li et al.

Column generation (CG) is a well-established method for solving large-scale linear programs. It involves iteratively optimizing a subproblem containing a subset of columns and using its dual solution to generate new columns with negative reduced costs. This process continues until the dual values converge to the optimal dual solution to the original problem. A natural phenomenon in CG is the heavy oscillation of the dual values during iterations, which can lead to a substantial slowdown in the convergence rate. Stabilization techniques are devised to accelerate the convergence of dual values by using information beyond the state of the current subproblem. However, there remains a significant gap in obtaining more accurate dual values at an earlier stage. To further narrow this gap, this paper introduces a novel approach consisting of 1) a machine learning approach for accurate prediction of optimal dual solutions and 2) an adaptive stabilization technique that effectively capitalizes on accurate predictions. On the graph coloring problem, we show that our method achieves a significantly improved convergence rate compared to traditional methods.

Xia Jiang, Yaoxin Wu, Minshuo Li et al.

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 generation 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.

Qi Li, Zhiguang Cao, Yining Ma et al.

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 $1.3\times$ to $15\times$ faster. Furthermore, we demonstrate that our method can also be applied to the Capacitated Vehicle Routing Problem (CVRP).

Ni Zhang, Zhiguang Cao, Jianan Zhou et al.

Complex vehicle routing problems (VRPs) remain a fundamental challenge, demanding substantial expert effort for intent interpretation and algorithm design. While large language models (LLMs) offer a promising path toward automation, current approaches still rely on external intervention, which restrict autonomy and often lead to execution errors and low solution feasibility. To address these challenges, we propose an Agentic Framework with LLMs (AFL) for solving complex vehicle routing problems, achieving full automation from problem instance to solution. AFL directly extracts knowledge from raw inputs and enables self-contained code generation without handcrafted modules or external solvers. To improve trustworthiness, AFL decomposes the overall pipeline into three manageable subtasks and employs four specialized agents whose coordinated interactions enforce cross-functional consistency and logical soundness. Extensive experiments on 60 complex VRPs, ranging from standard benchmarks to practical variants, validate the effectiveness and generality of our framework, showing comparable performance against meticulously designed algorithms. Notably, it substantially outperforms existing LLM-based baselines in both code reliability and solution feasibility, achieving rates close to 100% on the evaluated benchmarks.

Shiyan Liu, Bohan Tan, Zhiguang Cao et al.

Graph routing problems play a vital role in web-related networks, where finding optimal paths across graphs is essential for efficient data transmission and content delivery. Classic routing formulations such as the Traveling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) represent fundamental graph optimization challenges. Recent data-driven optimization methods have made significant progress, yet they often face limitations in training efficiency and generalization to large-scale instances. In this paper, we propose a novel Vision-augmented Asymmetric Group Preference Optimization (VAGPO) approach. By leveraging ResNet-based visual encoding and Transformer-based sequential modeling, VAGPO captures both spatial structure and temporal dependencies. Furthermore, we introduce an asymmetric group preference optimization strategy that significantly accelerates convergence compared to commonly used policy gradient methods. Experimental results on generated TSP and CVRP instances, as well as real-world datasets, demonstrate that the proposed VAGPO approach achieves highly competitive solution quality. Additionally, VAGPO exhibits strong generalization to larger instances (up to 1000 nodes) without re-training, highlighting its effectiveness in both learning efficiency and scalability.

Kuan Xu, Zhiguang Cao, Chenlong Zheng et al.

In this study, we propose a reinforcement learning-based adaptive variable neighborhood search (RL-AVNS) method designed for effectively solving the Vehicle Routing Problem with Multiple Time Windows (VRPMTW). Unlike traditional adaptive approaches that rely solely on historical operator performance, our method integrates a reinforcement learning framework to dynamically select neighborhood operators based on real-time solution states and learned experience. We introduce a fitness metric that quantifies customers' temporal flexibility to improve the shaking phase, and employ a transformer-based neural policy network to intelligently guide operator selection during the local search. Extensive computational experiments are conducted on realistic scenarios derived from the replenishment of unmanned vending machines, characterized by multiple clustered replenishment windows. Results demonstrate that RL-AVNS significantly outperforms traditional variable neighborhood search (VNS), adaptive VNS (AVNS), and state-of-the-art learning-based heuristics, achieving substantial improvements in solution quality and computational efficiency across various instance scales and time window complexities. Particularly notable is the algorithm's capability to generalize effectively to problem instances not encountered during training, underscoring its practical utility for complex logistics scenarios.

Yuan Jiang, Yaoxin Wu, Zhiguang Cao et al.

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.

Yaoxin Wu, Wen Song, Zhiguang Cao et al.

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.

Liang Xin, Wen Song, Zhiguang Cao et al.

We present NeuroLKH, a novel algorithm that combines deep learning with the strong traditional heuristic Lin-Kernighan-Helsgaun (LKH) for solving Traveling Salesman Problem. Specifically, we train a Sparse Graph Network (SGN) with supervised learning for edge scores and unsupervised learning for node penalties, both of which are critical for improving the performance of LKH. Based on the output of SGN, NeuroLKH creates the edge candidate set and transforms edge distances to guide the searching process of LKH. Extensive experiments firmly demonstrate that, by training one model on a wide range of problem sizes, NeuroLKH significantly outperforms LKH and generalizes well to much larger sizes. Also, we show that NeuroLKH can be applied to other routing problems such as Capacitated Vehicle Routing Problem (CVRP), Pickup and Delivery Problem (PDP), and CVRP with Time Windows (CVRPTW).

Jingwen Li, Liang Xin, Zhiguang Cao et al.

Recently, there is an emerging trend to apply deep reinforcement learning to solve the vehicle routing problem (VRP), where a learnt policy governs the selection of next node for visiting. However, existing methods could not handle well the pairing and precedence relationships in the pickup and delivery problem (PDP), which is a representative variant of VRP. To address this challenging issue, we leverage a novel neural network integrated with a heterogeneous attention mechanism to empower the policy in deep reinforcement learning to automatically select the nodes. In particular, the heterogeneous attention mechanism specifically prescribes attentions for each role of the nodes while taking into account the precedence constraint, i.e., the pickup node must precede the pairing delivery node. Further integrated with a masking scheme, the learnt policy is expected to find higher-quality solutions for solving PDP. Extensive experimental results show that our method outperforms the state-of-the-art heuristic and deep learning model, respectively, and generalizes well to different distributions and problem sizes.

Jingwen Li, Yining Ma, Ruize Gao et al.

Existing deep reinforcement learning (DRL) based methods for solving the capacitated vehicle routing problem (CVRP) intrinsically cope with homogeneous vehicle fleet, in which the fleet is assumed as repetitions of a single vehicle. Hence, their key to construct a solution solely lies in the selection of the next node (customer) to visit excluding the selection of vehicle. However, vehicles in real-world scenarios are likely to be heterogeneous with different characteristics that affect their capacity (or travel speed), rendering existing DRL methods less effective. In this paper, we tackle heterogeneous CVRP (HCVRP), where vehicles are mainly characterized by different capacities. We consider both min-max and min-sum objectives for HCVRP, which aim to minimize the longest or total travel time of the vehicle(s) in the fleet. To solve those problems, we propose a DRL method based on the attention mechanism with a vehicle selection decoder accounting for the heterogeneous fleet constraint and a node selection decoder accounting for the route construction, which learns to construct a solution by automatically selecting both a vehicle and a node for this vehicle at each step. Experimental results based on randomly generated instances show that, with desirable generalization to various problem sizes, our method outperforms the state-of-the-art DRL method and most of the conventional heuristics, and also delivers competitive performance against the state-of-the-art heuristic method, i.e., SISR. Additionally, the results of extended experiments demonstrate that our method is also able to solve CVRPLib instances with satisfactory performance.

Yining Ma, Jingwen Li, Zhiguang Cao et al.

Recently, Transformer has become a prevailing deep architecture for solving vehicle routing problems (VRPs). However, it is less effective in learning improvement models for VRP because its positional encoding (PE) method is not suitable in representing VRP solutions. This paper presents a novel Dual-Aspect Collaborative Transformer (DACT) to learn embeddings for the node and positional features separately, instead of fusing them together as done in existing ones, so as to avoid potential noises and incompatible correlations. Moreover, the positional features are embedded through a novel cyclic positional encoding (CPE) method to allow Transformer to effectively capture the circularity and symmetry of VRP solutions (i.e., cyclic sequences). We train DACT using Proximal Policy Optimization and design a curriculum learning strategy for better sample efficiency. We apply DACT to solve the traveling salesman problem (TSP) and capacitated vehicle routing problem (CVRP). Results show that our DACT outperforms existing Transformer based improvement models, and exhibits much better generalization performance across different problem sizes on synthetic and benchmark instances, respectively.

Liang Xin, Wen Song, Zhiguang Cao et al.

We present a novel deep reinforcement learning method to learn construction heuristics for vehicle routing problems. In specific, we propose a Multi-Decoder Attention Model (MDAM) to train multiple diverse policies, which effectively increases the chance of finding good solutions compared with existing methods that train only one policy. A customized beam search strategy is designed to fully exploit the diversity of MDAM. In addition, we propose an Embedding Glimpse layer in MDAM based on the recursive nature of construction, which can improve the quality of each policy by providing more informative embeddings. Extensive experiments on six different routing problems show that our method significantly outperforms the state-of-the-art deep learning based models.

Cong Zhang, Wen Song, Zhiguang Cao et al.

Priority dispatching rule (PDR) is widely used for solving real-world Job-shop scheduling problem (JSSP). However, the design of effective PDRs is a tedious task, requiring a myriad of specialized knowledge and often delivering limited performance. In this paper, we propose to automatically learn PDRs via an end-to-end deep reinforcement learning agent. We exploit the disjunctive graph representation of JSSP, and propose a Graph Neural Network based scheme to embed the states encountered during solving. The resulting policy network is size-agnostic, effectively enabling generalization on large-scale instances. Experiments show that the agent can learn high-quality PDRs from scratch with elementary raw features, and demonstrates strong performance against the best existing PDRs. The learned policies also perform well on much larger instances that are unseen in training.

Wen Song, Zhiguang Cao, Jie Zhang et al.

Backtracking search algorithms are often used to solve the Constraint Satisfaction Problem (CSP). The efficiency of backtracking search depends greatly on the variable ordering heuristics. Currently, the most commonly used heuristics are hand-crafted based on expert knowledge. In this paper, we propose a deep reinforcement learning based approach to automatically discover new variable ordering heuristics that are better adapted for a given class of CSP instances. We show that directly optimizing the search cost is hard for bootstrapping, and propose to optimize the expected cost of reaching a leaf node in the search tree. To capture the complex relations among the variables and constraints, we design a representation scheme based on Graph Neural Network that can process CSP instances with different sizes and constraint arities. Experimental results on random CSP instances show that the learned policies outperform classical hand-crafted heuristics in terms of minimizing the search tree size, and can effectively generalize to instances that are larger than those used in training.

Yaoxin Wu, Wen Song, Zhiguang Cao et al.

Recent studies in using deep learning to solve routing problems focus on construction heuristics, the solutions of which 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 hand-crafted rules which may limit their performance. In this paper, 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 next solution. We apply our method to two important routing problems, i.e. travelling salesman problem (TSP) and capacitated vehicle routing problem (CVRP). Experiments show that our method outperforms state-of-the-art deep learning based approaches. The learned policies are more effective than the traditional hand-crafted 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 dataset.