Zhenkun Wang

Fu Luo, Xi Lin, Fei Liu et al.

Neural combinatorial optimization (NCO) is a promising learning-based approach for solving challenging combinatorial optimization problems without specialized algorithm design by experts. However, most constructive NCO methods cannot solve problems with large-scale instance sizes, which significantly diminishes their usefulness for real-world applications. In this work, we propose a novel Light Encoder and Heavy Decoder (LEHD) model with a strong generalization ability to address this critical issue. The LEHD model can learn to dynamically capture the relationships between all available nodes of varying sizes, which is beneficial for model generalization to problems of various scales. Moreover, we develop a data-efficient training scheme and a flexible solution construction mechanism for the proposed LEHD model. By training on small-scale problem instances, the LEHD model can generate nearly optimal solutions for the Travelling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) with up to 1000 nodes, and also generalizes well to solve real-world TSPLib and CVRPLib problems. These results confirm our proposed LEHD model can significantly improve the state-of-the-art performance for constructive NCO. The code is available at https://github.com/CIAM-Group/NCO_code/tree/main/single_objective/LEHD.

Fei Liu, Xi Lin, Zhenkun Wang et al.

Multiobjective evolutionary algorithms (MOEAs) are major methods for solving multiobjective optimization problems (MOPs). Many MOEAs have been proposed in the past decades, of which the search operators need a carefully handcrafted design with domain knowledge. Recently, some attempts have been made to replace the manually designed operators in MOEAs with learning-based operators (e.g., neural network models). However, much effort is still required for designing and training such models, and the learned operators might not generalize well on new problems. To tackle the above challenges, this work investigates a novel approach that leverages the powerful large language model (LLM) to design MOEA operators. With proper prompt engineering, we successfully let a general LLM serve as a black-box search operator for decomposition-based MOEA (MOEA/D) in a zero-shot manner. In addition, by learning from the LLM behavior, we further design an explicit white-box operator with randomness and propose a new version of decomposition-based MOEA, termed MOEA/D-LO. Experimental studies on different test benchmarks show that our proposed method can achieve competitive performance with widely used MOEAs. It is also promising to see the operator only learned from a few instances can have robust generalization performance on unseen problems with quite different patterns and settings. The results reveal the potential benefits of using pre-trained LLMs in the design of MOEAs.To foster reproducibility and accessibility, the source code is https://github.com/FeiLiu36/LLM4MOEA.

Rui Zhang, Fei Liu, Xi Lin et al.

Automated heuristic design (AHD) has gained considerable attention for its potential to automate the development of effective heuristics. The recent advent of large language models (LLMs) has paved a new avenue for AHD, with initial efforts focusing on framing AHD as an evolutionary program search (EPS) problem. However, inconsistent benchmark settings, inadequate baselines, and a lack of detailed component analysis have left the necessity of integrating LLMs with search strategies and the true progress achieved by existing LLM-based EPS methods to be inadequately justified. This work seeks to fulfill these research queries by conducting a large-scale benchmark comprising four LLM-based EPS methods and four AHD problems across nine LLMs and five independent runs. Our extensive experiments yield meaningful insights, providing empirical grounding for the importance of evolutionary search in LLM-based AHD approaches, while also contributing to the advancement of future EPS algorithmic development. To foster accessibility and reproducibility, we have fully open-sourced our benchmark and corresponding results.

Chinedu Innocent Nwoye, Tong Yu, Saurav Sharma et al.

Formalizing surgical activities as triplets of the used instruments, actions performed, and target anatomies is becoming a gold standard approach for surgical activity modeling. The benefit is that this formalization helps to obtain a more detailed understanding of tool-tissue interaction which can be used to develop better Artificial Intelligence assistance for image-guided surgery. Earlier efforts and the CholecTriplet challenge introduced in 2021 have put together techniques aimed at recognizing these triplets from surgical footage. Estimating also the spatial locations of the triplets would offer a more precise intraoperative context-aware decision support for computer-assisted intervention. This paper presents the CholecTriplet2022 challenge, which extends surgical action triplet modeling from recognition to detection. It includes weakly-supervised bounding box localization of every visible surgical instrument (or tool), as the key actors, and the modeling of each tool-activity in the form of <instrument, verb, target> triplet. The paper describes a baseline method and 10 new deep learning algorithms presented at the challenge to solve the task. It also provides thorough methodological comparisons of the methods, an in-depth analysis of the obtained results across multiple metrics, visual and procedural challenges; their significance, and useful insights for future research directions and applications in surgery.

Shunyu Yao, Fei Liu, Xi Lin et al.

Heuristics are commonly used to tackle various search and optimization problems. Design heuristics usually require tedious manual crafting with domain knowledge. Recent works have incorporated Large Language Models (LLMs) into automatic heuristic search, leveraging their powerful language and coding capacity. However, existing research focuses on the optimal performance on the target problem as the sole objective, neglecting other criteria such as efficiency and scalability, which are vital in practice. To tackle this challenge, we propose to model the heuristic search as a multi-objective optimization problem and consider introducing additional practical criteria beyond optimal performance. Due to the complexity of the search space, conventional multi-objective optimization methods struggle to effectively handle LLM-based multi-objective heuristic search. We propose the first LLM-based multi-objective heuristic search framework, Multi-objective Evolution of Heuristic (MEoH), which integrates LLMs in a zero-shot manner to generate a non-dominated set of heuristics to meet multiple design criteria. We design a new dominance-dissimilarity mechanism for effective population management and selection, which incorporates both code dissimilarity in the search space and dominance in the objective space. MEoH is demonstrated in two well-known combinatorial optimization problems: the online Bin Packing Problem (BPP) and the Traveling Salesman Problem (TSP). The results indicate that a variety of elite heuristics are automatically generated in a single run, offering more trade-off options than the existing methods. It successfully achieves competitive or superior performance while improving efficiency up to 10 times. Moreover, we also observe that the multi-objective search introduces novel insights into heuristic design and leads to the discovery of diverse heuristics.

Rongsheng Chen, Changliang Zhou, Canhong Yu et al.

Generalist neural routing solvers have shown great potential in solving diverse vehicle routing problems (VRPs) with a unified model. However, existing solvers are typically limited to symmetric settings or degrade in performance when switching to asymmetric settings due to input inconsistencies or inherent structural differences, substantially limiting their practicality in real-world scenarios that encompass both scenarios. To address this limitation, we define the spatial position of each node based on the relative distances to a specific set of pivots and further propose a Spatial Pivot-Aligned Coordinate-free Embedding (SPACE) framework that unifies node representation and solution generation across symmetric and asymmetric VRPs. Specifically, we construct a bidirectional Frechet representation using a novel furthest pivot sampling strategy to enable invariant node representations across distinct problem settings. Furthermore, we introduce a weight-decomposed adaptive decoding mechanism that decouples geometric perception from problem representations, mitigating the overfitting of constraint decisions to a specific geometry setting. Extensive experiments on 110 VRP variants, comprising 55 symmetric problems and their asymmetric counterparts, demonstrate that SPACE achieves promising zero-shot generalization in both symmetric and asymmetric VRPs.

Heng Ping, Peiyu Zhang, Zhenkun Wang et al.

Applying large language models (LLMs) to RTL code optimization for improved power, performance, and area (PPA) faces two key challenges: ensuring functional correctness of optimized designs despite LLM hallucination, and systematically prioritizing power reduction within the multi-objective PPA trade-off space. We propose POET (Power-Oriented Evolutionary Tuning), a framework that addresses both challenges. For functional correctness, POET introduces a differential-testing-based testbench generation pipeline that treats the original design as a functional oracle, using deterministic simulation to produce golden references and eliminating LLM hallucination from the verification process. For PPA optimization, POET employs an LLM-driven evolutionary mechanism with non-dominated sorting, power-first intra-level ranking, and proportional survivor selection to steer the search toward the low-power region of the Pareto front without manual weight tuning. Evaluated on the RTL-OPT benchmark across 40 diverse RTL designs, POET achieves 100% functional correctness, the best power on all 40 designs, and competitive area and delay improvements.

Fei Liu, Xi Lin, Zhenkun Wang et al.

Vehicle routing problems (VRPs), which can be found in numerous real-world applications, have been an important research topic for several decades. Recently, the neural combinatorial optimization (NCO) approach that leverages a learning-based model to solve VRPs without manual algorithm design has gained substantial attention. However, current NCO methods typically require building one model for each routing problem, which significantly hinders their practical application for real-world industry problems with diverse attributes. In this work, we make the first attempt to tackle the crucial challenge of cross-problem generalization. In particular, we formulate VRPs as different combinations of a set of shared underlying attributes and solve them simultaneously via a single model through attribute composition. In this way, our proposed model can successfully solve VRPs with unseen attribute combinations in a zero-shot generalization manner. Extensive experiments are conducted on eleven VRP variants, benchmark datasets, and industry logistic scenarios. The results show that the unified model demonstrates superior performance in the eleven VRPs, reducing the average gap to around 5% from over 20% in the existing approach and achieving a significant performance boost on benchmark datasets as well as a real-world logistics application. The source code is included in https://github.com/FeiLiu36/MTNCO.

Yunpeng Ba, Xi Lin, Changliang Zhou et al.

Neural routing solvers (NRSs) that leverage deep learning to tackle vehicle routing problems have demonstrated notable potential for practical applications. By learning implicit heuristic rules from data, NRSs replace the handcrafted counterparts in classic heuristic frameworks, thereby reducing reliance on costly manual design and trial-and-error adjustments. This survey makes two main contributions: (1) The heuristic nature of NRSs is highlighted, and existing NRSs are reviewed from the perspective of heuristics. A hierarchical taxonomy based on heuristic principles is further introduced. (2) A generalization-focused evaluation pipeline is proposed to address limitations of the conventional pipeline. Comparative benchmarking of representative NRSs across both pipelines uncovers a series of previously unreported gaps in current research.

Zhi Zheng, Zhuoliang Xie, Zhenkun Wang et al.

Handcrafting heuristics for solving complex optimization tasks (e.g., route planning and task allocation) is a common practice but requires extensive domain knowledge. Recently, Large Language Model (LLM)-based automatic heuristic design (AHD) methods have shown promise in generating high-quality heuristics without manual interventions. Existing LLM-based AHD methods employ a population to maintain a fixed number of top-performing LLM-generated heuristics and introduce evolutionary computation (EC) to iteratively enhance the population. However, these population-based procedures cannot fully develop the potential of each heuristic and are prone to converge into local optima. To more comprehensively explore the space of heuristics, this paper proposes to use Monte Carlo Tree Search (MCTS) for LLM-based heuristic evolution. The proposed MCTS-AHD method organizes all LLM-generated heuristics in a tree structure and can better develop the potential of temporarily underperforming heuristics. In experiments, MCTS-AHD delivers significantly higher-quality heuristics on various complex tasks. Our code is available.

Zhongju Yuan, Zhenkun Wang, Genghui Li

Few-shot relation extraction aims to recognize novel relations with few labeled sentences in each relation. Previous metric-based few-shot relation extraction algorithms identify relationships by comparing the prototypes generated by the few labeled sentences embedding with the embeddings of the query sentences using a trained metric function. However, as these domains always have considerable differences from those in the training dataset, the generalization ability of these approaches on unseen relations in many domains is limited. Since the prototype is necessary for obtaining relationships between entities in the latent space, we suggest learning more interpretable and efficient prototypes from prior knowledge and the intrinsic semantics of relations to extract new relations in various domains more effectively. By exploring the relationships between relations using prior information, we effectively improve the prototype representation of relations. By using contrastive learning to make the classification margins between sentence embedding more distinct, the prototype's geometric interpretability is enhanced. Additionally, utilizing a transfer learning approach for the cross-domain problem allows the generation process of the prototype to account for the gap between other domains, making the prototype more robust and enabling the better extraction of associations across multiple domains. The experiment results on the benchmark FewRel dataset demonstrate the advantages of the suggested method over some state-of-the-art approaches.

Zhuoliang Xie, Fei Liu, Zhenkun Wang et al.

The Capacitated Vehicle Routing Problem (CVRP), a fundamental combinatorial optimization challenge, focuses on optimizing fleet operations under vehicle capacity constraints. While extensively studied in operational research, the NP-hard nature of CVRP continues to pose significant computational challenges, particularly for large-scale instances. This study presents AILS-AHD (Adaptive Iterated Local Search with Automatic Heuristic Design), a novel approach that leverages Large Language Models (LLMs) to revolutionize CVRP solving. Our methodology integrates an evolutionary search framework with LLMs to dynamically generate and optimize ruin heuristics within the AILS method. Additionally, we introduce an LLM-based acceleration mechanism to enhance computational efficiency. Comprehensive experimental evaluations against state-of-the-art solvers, including AILS-II and HGS, demonstrate the superior performance of AILS-AHD across both moderate and large-scale instances. Notably, our approach establishes new best-known solutions for 8 out of 10 instances in the CVRPLib large-scale benchmark, underscoring the potential of LLM-driven heuristic design in advancing the field of vehicle routing optimization.

Anzhe Cheng, Zhenkun Wang, Chenzhong Yin et al.

Backpropagation (BP) has been a successful optimization technique for deep learning models. However, its limitations, such as backward- and update-locking, and its biological implausibility, hinder the concurrent updating of layers and do not mimic the local learning processes observed in the human brain. To address these issues, recent research has suggested using local error signals to asynchronously train network blocks. However, this approach often involves extensive trial-and-error iterations to determine the best configuration for local training. This includes decisions on how to decouple network blocks and which auxiliary networks to use for each block. In our work, we introduce a novel BP-free approach: a block-wise BP-free (BWBPF) neural network that leverages local error signals to optimize distinct sub-neural networks separately, where the global loss is only responsible for updating the output layer. The local error signals used in the BP-free model can be computed in parallel, enabling a potential speed-up in the weight update process through parallel implementation. Our experimental results consistently show that this approach can identify transferable decoupled architectures for VGG and ResNet variations, outperforming models trained with end-to-end backpropagation and other state-of-the-art block-wise learning techniques on datasets such as CIFAR-10 and Tiny-ImageNet. The code is released at https://github.com/Belis0811/BWBPF.

Fei Liu, Xi Lin, Weiduo Liao et al.

Neural combinatorial optimization (NCO) is a promising learning-based approach to solving various vehicle routing problems without much manual algorithm design. However, the current NCO methods mainly focus on the in-distribution performance, while the real-world problem instances usually come from different distributions. A costly fine-tuning approach or generalized model retraining from scratch could be needed to tackle the out-of-distribution instances. Unlike the existing methods, this work investigates an efficient prompt learning approach in NCO for cross-distribution adaptation. To be concrete, we propose a novel prompt learning method to facilitate fast zero-shot adaptation of a pre-trained model to solve routing problem instances from different distributions. The proposed model learns a set of prompts among various distributions and then selects the best-matched one to prompt a pre-trained attention model for each problem instance. Extensive experiments show that the proposed prompt learning approach facilitates the fast adaptation of pre-trained routing models. It also outperforms existing generalized models on both in-distribution prediction and zero-shot generalization to a diverse set of new tasks. Our code implementation is available online https://github.com/FeiLiu36/PromptVRP.

Tianqi Xiang, Wenjun Yue, Yiqun Lin et al.

Performing magnetic resonance imaging (MRI) reconstruction from under-sampled k-space data can accelerate the procedure to acquire MRI scans and reduce patients' discomfort. The reconstruction problem is usually formulated as a denoising task that removes the noise in under-sampled MRI image slices. Although previous GAN-based methods have achieved good performance in image denoising, they are difficult to train and require careful tuning of hyperparameters. In this paper, we propose a novel MRI denoising framework DiffCMR by leveraging conditional denoising diffusion probabilistic models. Specifically, DiffCMR perceives conditioning signals from the under-sampled MRI image slice and generates its corresponding fully-sampled MRI image slice. During inference, we adopt a multi-round ensembling strategy to stabilize the performance. We validate DiffCMR with cine reconstruction and T1/T2 mapping tasks on MICCAI 2023 Cardiac MRI Reconstruction Challenge (CMRxRecon) dataset. Results show that our method achieves state-of-the-art performance, exceeding previous methods by a significant margin. Code is available at https://github.com/xmed-lab/DiffCMR.

Lindong Xie, Genghui Li, Zhenkun Wang et al.

Surrogate-assisted evolutionary algorithms (SAEAs) are a key tool for addressing costly optimization tasks, with their efficiency being heavily dependent on the selection of surrogate models and infill sampling criteria. However, designing an effective dynamic selection strategy for SAEAs is labor-intensive and requires substantial domain knowledge. To address this challenge, this paper proposes LLM-SAEA, a novel approach that integrates large language models (LLMs) to configure both surrogate models and infill sampling criteria online. Specifically, LLM-SAEA develops a collaboration-of-experts framework, where one LLM serves as a scoring expert (LLM-SE), assigning scores to surrogate models and infill sampling criteria based on their optimization performance, while another LLM acts as a decision expert (LLM-DE), selecting the appropriate configurations by analyzing their scores along with the current optimization state. Experimental results demonstrate that LLM-SAEA outperforms several state-of-the-art algorithms across standard test cases. The source code is publicly available at https://github.com/ForrestXie9/LLM-SAEA.

Changliang Zhou, Xi Lin, Zhenkun Wang et al.

The neural combinatorial optimization (NCO) method has shown great potential for solving routing problems of intelligent transportation systems without requiring expert knowledge. However, existing constructive NCO methods still struggle to solve large-scale instances, which significantly limits their application prospects. To address these crucial shortcomings, this work proposes a novel Instance-Conditioned Adaptation Model (ICAM) for better large-scale generalization of neural routing solvers. In particular, we design a simple yet efficient instance-conditioned adaptation function to significantly improve the generalization performance of existing NCO models with a small time and memory overhead. In addition, with a systematic investigation on the performance of information incorporation between different attention mechanisms, we further propose a powerful yet low-complexity instance-conditioned adaptation module to generate better solutions for instances across different scales. Extensive experimental results on both synthetic and benchmark instances show that our proposed method is capable of obtaining promising results with a very fast inference time in solving large-scale Traveling Salesman Problems (TSPs), Capacitated Vehicle Routing Problems (CVRPs), and Asymmetric Traveling Salesman Problems (ATSPs). Our code is available at https://github.com/CIAM-Group/ICAM.

Canhong Yu, Changliang Zhou, Rongsheng Chen et al.

Heavy-Encoder-Light-Decoder (HELD) neural routing solvers have emerged as a promising paradigm due to their broad applicability across multiple vehicle routing problems (VRPs). However, they typically struggle with VRP variants with complex constraints. To address this limitation, this paper systematically revisits existing neural solvers from the perspective of the generation mechanism for state embeddings (i.e., query vector prior to compatibility calculation) during decoding. We identify that current mechanisms restrict the observation space during attention computation, introducing a key bottleneck to achieving high-quality solutions. Through detailed empirical analysis, we demonstrate the necessity of preserving a global observation space. To overcome the constraint-agnostic drawback inherent to global observation spaces, we propose a simple yet powerful Constraint-Aware Residual Modulation (CARM) module. By adaptively modulating the context embedding with constraint-relevant variables, CARM effectively enhances constraint awareness, enabling the neural solver to fully leverage the global observation space and generate an efficient state embedding. Extensive experimental results across two single-task and five multi-task neural routing solvers confirm that the CARM module consistently boosts baseline performance. Notably, solvers equipped with our CARM achieve substantial improvements in scaling to large-scale instances and in generalizing to unseen VRP variants. These findings provide valuable insights for the architectural design of neural routing solvers.

Fei Liu, Xialiang Tong, Mingxuan Yuan et al.

Heuristics are widely used for dealing with complex search and optimization problems. However, manual design of heuristics can be often very labour extensive and requires rich working experience and knowledge. This paper proposes Evolution of Heuristic (EoH), a novel evolutionary paradigm that leverages both Large Language Models (LLMs) and Evolutionary Computation (EC) methods for Automatic Heuristic Design (AHD). EoH represents the ideas of heuristics in natural language, termed thoughts. They are then translated into executable codes by LLMs. The evolution of both thoughts and codes in an evolutionary search framework makes it very effective and efficient for generating high-performance heuristics. Experiments on three widely studied combinatorial optimization benchmark problems demonstrate that EoH outperforms commonly used handcrafted heuristics and other recent AHD methods including FunSearch. Particularly, the heuristic produced by EoH with a low computational budget (in terms of the number of queries to LLMs) significantly outperforms widely-used human hand-crafted baseline algorithms for the online bin packing problem.

Zhi Zheng, Changliang Zhou, Tong Xialiang et al.

Single-stage neural combinatorial optimization solvers have achieved near-optimal results on various small-scale combinatorial optimization (CO) problems without requiring expert knowledge. However, these solvers exhibit significant performance degradation when applied to large-scale CO problems. Recently, two-stage neural methods motivated by divide-and-conquer strategies have shown efficiency in addressing large-scale CO problems. Nevertheless, the performance of these methods highly relies on problem-specific heuristics in either the dividing or the conquering procedure, which limits their applicability to general CO problems. Moreover, these methods employ separate training schemes and ignore the interdependencies between the dividing and conquering strategies, often leading to sub-optimal solutions. To tackle these drawbacks, this article develops a unified neural divide-and-conquer framework (i.e., UDC) for solving general large-scale CO problems. UDC offers a Divide-Conquer-Reunion (DCR) training method to eliminate the negative impact of a sub-optimal dividing policy. Employing a high-efficiency Graph Neural Network (GNN) for global instance dividing and a fixed-length sub-path solver for conquering divided sub-problems, the proposed UDC framework demonstrates extensive applicability, achieving superior performance in 10 representative large-scale CO problems. The code is available at https://github.com/CIAM-Group/NCO_code/tree/main/single_objective/UDC-Large-scale-CO-master.

Ping Guo, Cheng Gong, Xi Lin et al.

Crafting adversarial examples is crucial for evaluating and enhancing the robustness of Deep Neural Networks (DNNs), presenting a challenge equivalent to maximizing a non-differentiable 0-1 loss function. However, existing single objective methods, namely adversarial attacks focus on a surrogate loss function, do not fully harness the benefits of engaging multiple loss functions, as a result of insufficient understanding of their synergistic and conflicting nature. To overcome these limitations, we propose the Multi-Objective Set-based Attack (MOS Attack), a novel adversarial attack framework leveraging multiple loss functions and automatically uncovering their interrelations. The MOS Attack adopts a set-based multi-objective optimization strategy, enabling the incorporation of numerous loss functions without additional parameters. It also automatically mines synergistic patterns among various losses, facilitating the generation of potent adversarial attacks with fewer objectives. Extensive experiments have shown that our MOS Attack outperforms single-objective attacks. Furthermore, by harnessing the identified synergistic patterns, MOS Attack continues to show superior results with a reduced number of loss functions. Our code is available at https://github.com/pgg3/MOS-Attack.

Fei Liu, Yiming Yao, Ping Guo et al.

Algorithm Design (AD) is crucial for effective problem-solving across various domains. The advent of Large Language Models (LLMs) has notably enhanced the automation and innovation within this field, offering new perspectives and promising solutions. Over the past three years, the integration of LLMs into AD (LLM4AD) has seen substantial progress, with applications spanning optimization, machine learning, mathematical reasoning, and scientific discovery. Given the rapid advancements and expanding scope of this field, a systematic review is both timely and necessary. This paper provides a systematic review of LLM4AD. First, we offer an overview and summary of existing studies. Then, we introduce a taxonomy and review the literature across four dimensions: the roles of LLMs, search methods, prompt methods, and application domains with a discussion of potential and achievements of LLMs in AD. Finally, we identify current challenges and highlight several promising directions for future research.

Xi Lin, Xiaoyuan Zhang, Zhiyuan Yang et al.

Multi-objective optimization problems can be found in many real-world applications, where the objectives often conflict each other and cannot be optimized by a single solution. In the past few decades, numerous methods have been proposed to find Pareto solutions that represent optimal trade-offs among the objectives for a given problem. However, these existing methods could have high computational complexity or may not have good theoretical properties for solving a general differentiable multi-objective optimization problem. In this work, by leveraging the smooth optimization technique, we propose a lightweight and efficient smooth Tchebycheff scalarization approach for gradient-based multi-objective optimization. It has good theoretical properties for finding all Pareto solutions with valid trade-off preferences, while enjoying significantly lower computational complexity compared to other methods. Experimental results on various real-world application problems fully demonstrate the effectiveness of our proposed method.

Fu Luo, Xi Lin, Zhenkun Wang et al.

The end-to-end neural combinatorial optimization (NCO) method shows promising performance in solving complex combinatorial optimization problems without the need for expert design. However, existing methods struggle with large-scale problems, hindering their practical applicability. To overcome this limitation, this work proposes a novel Self-Improved Learning (SIL) method for better scalability of neural combinatorial optimization. Specifically, we develop an efficient self-improved mechanism that enables direct model training on large-scale problem instances without any labeled data. Powered by an innovative local reconstruction approach, this method can iteratively generate better solutions by itself as pseudo-labels to guide efficient model training. In addition, we design a linear complexity attention mechanism for the model to efficiently handle large-scale combinatorial problem instances with low computation overhead. Comprehensive experiments on the Travelling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) with up to 100K nodes in both uniform and real-world distributions demonstrate the superior scalability of our method.

Fei Liu, Rui Zhang, Zhuoliang Xie et al.

We introduce LLM4AD, a unified Python platform for algorithm design (AD) with large language models (LLMs). LLM4AD is a generic framework with modularized blocks for search methods, algorithm design tasks, and LLM interface. The platform integrates numerous key methods and supports a wide range of algorithm design tasks across various domains including optimization, machine learning, and scientific discovery. We have also designed a unified evaluation sandbox to ensure a secure and robust assessment of algorithms. Additionally, we have compiled a comprehensive suite of support resources, including tutorials, examples, a user manual, online resources, and a dedicated graphical user interface (GUI) to enhance the usage of LLM4AD. We believe this platform will serve as a valuable tool for fostering future development in the merging research direction of LLM-assisted algorithm design.

Han Li, Fei Liu, Zhi Zheng et al.

Vehicle Routing Problems (VRPs) are significant Combinatorial Optimization (CO) problems holding substantial practical importance. Recently, Neural Combinatorial Optimization (NCO), which involves training deep learning models on extensive data to learn vehicle routing heuristics, has emerged as a promising approach due to its efficiency and the reduced need for manual algorithm design. However, applying NCO across diverse real-world scenarios with various constraints necessitates cross-problem capabilities. Current NCO methods typically employ a unified model lacking a constraint-specific structure, thereby restricting their cross-problem performance. Current multi-task methods for VRPs typically employ a constraint-unaware model, limiting their cross-problem performance. Furthermore, they rely solely on global connectivity, which fails to focus on key nodes and leads to inefficient representation learning. This paper introduces a Constraint-Aware Dual-Attention Model (CaDA), designed to address these limitations. CaDA incorporates a constraint prompt that efficiently represents different problem variants. Additionally, it features a dual-attention mechanism with a global branch for capturing broader graph-wide information and a sparse branch that selectively focuses on the most relevant nodes. We comprehensively evaluate our model on 16 different VRPs and compare its performance against existing cross-problem VRP solvers. CaDA achieves state-of-the-art results across all the VRPs. Our ablation study further confirms that each component of CaDA contributes positively to its cross-problem learning performance.

Changliang Zhou, Xi Lin, Zhenkun Wang et al.

Constructive neural combinatorial optimization (NCO) has attracted growing research attention due to its ability to solve complex routing problems without relying on handcrafted rules. However, existing NCO methods face significant challenges in generalizing to large-scale problems due to high computational complexity and inefficient capture of structural patterns. To address this issue, we propose a novel learning-based search space reduction method that adaptively selects a small set of promising candidate nodes at each step of the constructive NCO process. Unlike traditional methods that rely on fixed heuristics, our selection model dynamically prioritizes nodes based on learned patterns, significantly reducing the search space while maintaining solution quality. Experimental results demonstrate that our method, trained solely on 100-node instances from uniform distribution, generalizes remarkably well to large-scale Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) instances with up to 1 million nodes from the uniform distribution and over 80K nodes from other distributions.

Kai Li, Fei Liu, Zhenkun Wang et al.

Real-world Vehicle Routing Problems (VRPs) are characterized by a variety of practical constraints, making manual solver design both knowledge-intensive and time-consuming. Although there is increasing interest in automating the design of routing algorithms, existing research has explored only a limited array of VRP variants and fails to adequately address the complex and prevalent constraints encountered in real-world situations. To fill this gap, this paper introduces RoutBench, a benchmark of 1,000 VRP variants derived from 24 attributes, for evaluating the effectiveness of automatic routing solvers in addressing complex constraints. Along with RoutBench, we present the Automatic Routing Solver (ARS), which employs Large Language Model (LLM) agents to enhance a backbone algorithm framework by automatically generating constraint-aware heuristic code, based on problem descriptions and several representative constraints selected from a database. Our experiments show that ARS outperforms state-of-the-art LLM-based methods and commonly used solvers, automatically solving 91.67% of common VRPs and achieving at least a 30% improvement across all benchmarks.

Changliang Zhou, Canhong Yu, Shunyu Yao et al.

Multi-task neural routing solvers have emerged as a promising paradigm for their ability to solve multiple vehicle routing problems (VRPs) using a single model. However, existing neural solvers typically rely on predefined problem constraints or require per-problem fine-tuning, which substantially limits their zero-shot generalization ability to unseen VRP variants. To address this critical bottleneck, we propose URS, a unified neural routing solver capable of zero-shot generalization across a wide range of unseen VRPs using a single model without any fine-tuning. The key component of URS is the unified data representation (UDR), which replaces problem enumeration with data unification, thereby broadening the problem coverage and reducing reliance on domain expertise. In addition, we propose a Mixed Bias Module (MBM) to efficiently learn the geometric and relational biases inherent in various problems. On top of the proposed UDR, we further develop a parameter generator that adaptively adjusts the decoder and bias weights of MBM to enhance zero-shot generalization. Moreover, we propose an LLM-driven constraint satisfaction mechanism, which translates raw problem descriptions into executable stepwise masking functions to ensure solution feasibility. Extensive experiments demonstrate that URS can consistently produce high-quality solutions for more than 100 distinct VRP variants without any fine-tuning, which includes more than 90 unseen variants. To the best of our knowledge, URS is the first neural solver capable of handling over 100 VRP variants with a single model.

Kai Li, Ruihao Zheng, Xinye Hao et al.

In real-world routing problems, users often propose conflicting or unreasonable requirements, which result in infeasible optimization models due to overly restrictive or contradictory constraints, leading to an empty feasible solution set. Existing Large Language Model (LLM)-based methods attempt to diagnose infeasible models, but modifying such models often involves multiple potential adjustments that these methods do not consider. To fill this gap, we introduce Multi-Objective Infeasibility Diagnosis (MOID), which combines LLM agents and multi-objective optimization within an automatic routing solver, to provide a set of representative actionable suggestions. Specifically, MOID employs multi-objective optimization to consider both path cost and constraint violation, generating a set of trade-off solutions, each encompassing varying degrees of model adjustments. To extract practical insights from these solutions, MOID utilizes LLM agents to generate a solution analysis function for the infeasible model. This function analyzes these distinct solutions to diagnose the original infeasible model, providing users with diverse diagnostic insights and suggestions. Finally, we compare MOID with several LLM-based methods on 50 types of infeasible routing problems. The results indicate that MOID automatically generates multiple diagnostic suggestions in a single run, providing more practical insights for restoring model feasibility and decision-making compared to existing methods.

Fu Luo, Yaoxin Wu, Zhi Zheng et al.

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 proposes 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 performances on the CVRP and CVRP with time windows (CVRPTW) with various constraint tightness degrees.

Yuepeng Zheng, Fu Luo, Zhenkun Wang et al.

Multi-Task Learning (MTL) in Neural Combinatorial Optimization (NCO) is a promising approach to train 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 the 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 1000 nodes indicate that our proposed method consistently achieves superior performance on both uniform and real-world benchmarks, demonstrating robust generalization abilities.

Yuanyao Chen, Rongsheng Chen, Fu Luo et al.

Neural Combinatorial Optimization (NCO) has emerged as a promising learning-based paradigm for addressing Vehicle Routing Problems (VRPs) by minimizing the need for extensive manual engineering. While existing NCO methods, trained on small-scale instances (e.g., 100 nodes), have demonstrated considerable success on problems of similar scale, their performance significantly degrades when applied to large-scale scenarios. This degradation arises from the distributional shift between training and testing data, rendering policies learned on small instances ineffective for larger problems. To overcome this limitation, we introduce a novel learning framework driven by Large Language Models (LLMs). This framework learns a projection between the training and testing distributions, which is then deployed to enhance the scalability of the NCO model. Notably, unlike prevailing techniques that necessitate joint training with the neural network, our approach operates exclusively during the inference phase, obviating the need for model retraining. Extensive experiments demonstrate that our method enables a backbone model (trained on 100-node instances) to achieve superior performance on large-scale Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) of up to 100K nodes from diverse distributions.

Fu Luo, Xi Lin, Mengyuan Zhong et al.

Neural Combinatorial Optimisation (NCO) is a promising learning-based approach for solving Vehicle Routing Problems (VRPs) without extensive manual design. While existing constructive NCO methods typically follow an appending-based paradigm that sequentially adds unvisited nodes to partial solutions, this rigid approach often leads to suboptimal results. To overcome this limitation, we explore the idea of insertion-based paradigm and propose Learning to Construct with Insertion-based Paradigm (L2C-Insert), a novel learning-based method for constructive NCO. Unlike traditional approaches, L2C-Insert builds solutions by strategically inserting unvisited nodes at any valid position in the current partial solution, which can significantly enhance the flexibility and solution quality. The proposed framework introduces three key components: a novel model architecture for precise insertion position prediction, an efficient training scheme for model optimization, and an advanced inference technique that fully exploits the insertion paradigm's flexibility. Extensive experiments on both synthetic and real-world instances of the Travelling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) demonstrate that L2C-Insert consistently achieves superior performance across various problem sizes.

Han Li, Fei Liu, Zhenkun Wang et al.

Vehicle routing problems (VRPs) are central to combinatorial optimization with significant practical implications. Recent advancements in neural combinatorial optimization (NCO) have demonstrated promising results by leveraging neural networks to solve VRPs, yet the exploration of model scaling within this domain remains underexplored. Inspired by the success of model scaling in large language models (LLMs), this study introduces a Large Routing Model with 1 billion parameters (LRM-1B), designed to address diverse VRP scenarios. We present a comprehensive evaluation of LRM-1B across multiple problem variants, distributions, and sizes, establishing state-of-the-art results. Our findings reveal that LRM-1B not only adapts to different VRP challenges but also showcases superior performance, outperforming existing models. Additionally, we explore the scaling behavior of neural routing models from 1M to 1B parameters. Our analysis confirms power-law between multiple model factors and performance, offering critical insights into the optimal configurations for foundation neural routing solvers.