GELD: A Unified Neural Model for Efficiently Solving Traveling Salesman Problems Across Different Scales

The Traveling Salesman Problem (TSP) is a well-known combinatorial optimization problem with broad real-world applications. Recent advancements in neural network-based TSP solvers have shown promising results. Nonetheless, these models often struggle to efficiently solve both small- and large-scale TSPs using the same set of pre-trained model parameters, limiting their practical utility. To address this issue, we introduce a novel neural TSP solver named GELD, built upon our proposed broad global assessment and refined local selection framework. Specifically, GELD integrates a lightweight Global-view Encoder (GE) with a heavyweight Local-view Decoder (LD) to enrich embedding representation while accelerating the decision-making process. Moreover, GE incorporates a novel low-complexity attention mechanism, allowing GELD to achieve low inference latency and scalability to larger-scale TSPs. Additionally, we propose a two-stage training strategy that utilizes training instances of different sizes to bolster GELD's generalization ability. Extensive experiments conducted on both synthetic and real-world datasets demonstrate that GELD outperforms seven state-of-the-art models considering both solution quality and inference speed. Furthermore, GELD can be employed as a post-processing method to significantly elevate the quality of the solutions derived by existing neural TSP solvers via spending affordable additional computing time. Notably, GELD is shown as capable of solving TSPs with up to 744,710 nodes, first-of-its-kind to solve this large size TSP without relying on divide-and-conquer strategies to the best of our knowledge.

Efficient Heuristics Generation for Solving Combinatorial Optimization Problems Using Large Language Models

Recent studies exploited Large Language Models (LLMs) to autonomously generate heuristics for solving Combinatorial Optimization Problems (COPs), by prompting LLMs to first provide search directions and then derive heuristics accordingly. However, the absence of task-specific knowledge in prompts often leads LLMs to provide unspecific search directions, obstructing the derivation of well-performing heuristics. Moreover, evaluating the derived heuristics remains resource-intensive, especially for those semantically equivalent ones, often requiring omissible resource expenditure. To enable LLMs to provide specific search directions, we propose the Hercules algorithm, which leverages our designed Core Abstraction Prompting (CAP) method to abstract the core components from elite heuristics and incorporate them as prior knowledge in prompts. We theoretically prove the effectiveness of CAP in reducing unspecificity and provide empirical results in this work. To reduce computing resources required for evaluating the derived heuristics, we propose few-shot Performance Prediction Prompting (PPP), a first-of-its-kind method for the Heuristic Generation (HG) task. PPP leverages LLMs to predict the fitness values of newly derived heuristics by analyzing their semantic similarity to previously evaluated ones. We further develop two tailored mechanisms for PPP to enhance predictive accuracy and determine unreliable predictions, respectively. The use of PPP makes Hercules more resource-efficient and we name this variant Hercules-P. Extensive experiments across four HG tasks, five COPs, and eight LLMs demonstrate that Hercules outperforms the state-of-the-art LLM-based HG algorithms, while Hercules-P excels at minimizing required computing resources. In addition, we illustrate the effectiveness of CAP, PPP, and the other proposed mechanisms by conducting relevant ablation studies.

Let Synthetic Data Shine: Domain Reassembly and Soft-Fusion for Single Domain Generalization

Single Domain Generalization (SDG) aims to train models with consistent performance across diverse scenarios using data from a single source. While using latent diffusion models (LDMs) show promise in augmenting limited source data, we demonstrate that directly using synthetic data can be detrimental due to significant feature distribution discrepancies between synthetic and real target domains, leading to performance degradation. To address this issue, we propose Discriminative Domain Reassembly and Soft-Fusion (DRSF), a training framework leveraging synthetic data to improve model generalization. We employ LDMs to produce diverse pseudo-target domain samples and introduce two key modules to handle distribution bias. First, Discriminative Feature Decoupling and Reassembly (DFDR) module uses entropy-guided attention to recalibrate channel-level features, suppressing synthetic noise while preserving semantic consistency. Second, Multi-pseudo-domain Soft Fusion (MDSF) module uses adversarial training with latent-space feature interpolation, creating continuous feature transitions between domains. Extensive SDG experiments on object detection and semantic segmentation tasks demonstrate that DRSF achieves substantial performance gains with only marginal computational overhead. Notably, DRSF's plug-and-play architecture enables seamless integration with unsupervised domain adaptation paradigms, underscoring its broad applicability in addressing diverse and real-world domain challenges.

An Efficient Diffusion-based Non-Autoregressive Solver for Traveling Salesman Problem

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 $\href{https://github.com/DEITSP/DEITSP}{https://github.com/DEITSP/DEITSP}$.

Improving Generalization of Neural Vehicle Routing Problem Solvers Through the Lens of Model Architecture

Neural models produce promising results when solving Vehicle Routing Problems (VRPs), but often fall short in generalization. Recent attempts to enhance model generalization often incur unnecessarily large training cost or cannot be directly applied to other models solving different VRP variants. To address these issues, we take a novel perspective on model architecture in this study. Specifically, we propose a plug-and-play Entropy-based Scaling Factor (ESF) and a Distribution-Specific (DS) decoder to enhance the size and distribution generalization, respectively. ESF adjusts the attention weight pattern of the model towards familiar ones discovered during training when solving VRPs of varying sizes. The DS decoder explicitly models VRPs of multiple training distribution patterns through multiple auxiliary light decoders, expanding the model representation space to encompass a broader range of distributional scenarios. We conduct extensive experiments on both synthetic and widely recognized real-world benchmarking datasets and compare the performance with seven baseline models. The results demonstrate the effectiveness of using ESF and DS decoder to obtain a more generalizable model and showcase their applicability to solve different VRP variants, i.e., travelling salesman problem and capacitated VRP. Notably, our proposed generic components require minimal computational resources, and can be effortlessly integrated into conventional generalization strategies to further elevate model generalization.

Neural Combinatorial Optimization Algorithms for Solving Vehicle Routing Problems: A Comprehensive Survey with Perspectives

Although several surveys on Neural Combinatorial Optimization (NCO) solvers specifically designed to solve Vehicle Routing Problems (VRPs) have been conducted. These existing surveys did not cover the state-of-the-art (SOTA) NCO solvers emerged recently. More importantly, to provide a comprehensive taxonomy of NCO solvers with up-to-date coverage, based on our thorough review of relevant publications and preprints, we divide all NCO solvers into four distinct categories, namely Learning to Construct, Learning to Improve, Learning to Predict-Once, and Learning to Predict-Multiplicity solvers. Subsequently, we present the inadequacies of the SOTA solvers, including poor generalization, incapability to solve large-scale VRPs, inability to address most types of VRP variants simultaneously, and difficulty in comparing these NCO solvers with the conventional Operations Research algorithms. Simultaneously, we propose promising and viable directions to overcome these inadequacies. In addition, we compare the performance of representative NCO solvers from the Reinforcement, Supervised, and Unsupervised Learning paradigms across both small- and large-scale VRPs. Finally, following the proposed taxonomy, we provide an accompanying web page as a live repository for NCO solvers. Through this survey and the live repository, we hope to make the research community of NCO solvers for VRPs more thriving.

Distilling Autoregressive Models to Obtain High-Performance Non-Autoregressive Solvers for Vehicle Routing Problems with Faster Inference Speed

Neural construction models have shown promising performance for Vehicle Routing Problems (VRPs) by adopting either the Autoregressive (AR) or Non-Autoregressive (NAR) learning approach. While AR models produce high-quality solutions, they generally have a high inference latency due to their sequential generation nature. Conversely, NAR models generate solutions in parallel with a low inference latency but generally exhibit inferior performance. In this paper, we propose a generic Guided Non-Autoregressive Knowledge Distillation (GNARKD) method to obtain high-performance NAR models having a low inference latency. GNARKD removes the constraint of sequential generation in AR models while preserving the learned pivotal components in the network architecture to obtain the corresponding NAR models through knowledge distillation. We evaluate GNARKD by applying it to three widely adopted AR models to obtain NAR VRP solvers for both synthesized and real-world instances. The experimental results demonstrate that GNARKD significantly reduces the inference time (4-5 times faster) with acceptable performance drop (2-3\%). To the best of our knowledge, this study is first-of-its-kind to obtain NAR VRP solvers from AR ones through knowledge distillation.

Reinforcement Learning-based Non-Autoregressive Solver for Traveling Salesman Problems

The Traveling Salesman Problem (TSP) is a well-known problem in combinatorial optimization with applications in various domains. However, existing TSP solvers face challenges in producing high-quality solutions with low latency. To address this issue, we propose NAR4TSP, which produces TSP solutions in a Non-Autoregressive (NAR) manner using a specially designed Graph Neural Network (GNN), achieving faster inference speed. Moreover, NAR4TSP is trained using an enhanced Reinforcement Learning (RL) strategy, eliminating the dependency on costly labels used to train conventional supervised learning-based NAR models. To the best of our knowledge, NAR4TSP is the first TSP solver that successfully combines RL and NAR decoding. The experimental results on both synthetic and real-world TSP instances demonstrate that NAR4TSP outperforms four state-of-the-art models in terms of solution quality, inference latency, and generalization ability. Lastly, we present visualizations of NAR4TSP's decoding process and its overall path planning to showcase the feasibility of implementing NAR4TSP in an end-to-end manner and its effectiveness, respectively.

Shape-Aware Fine-Grained Classification of Erythroid Cells

Fine-grained classification and counting of bone marrow erythroid cells are vital for evaluating the health status and formulating therapeutic schedules for leukemia or hematopathy. Due to the subtle visual differences between different types of erythroid cells, it is challenging to apply existing image-based deep learning models for fine-grained erythroid cell classification. Moreover, there is no large open-source datasets on erythroid cells to support the model training. In this paper, we introduce BMEC (Bone Morrow Erythroid Cells), the first large fine-grained image dataset of erythroid cells, to facilitate more deep learning research on erythroid cells. BMEC contains 5,666 images of individual erythroid cells, each of which is extracted from the bone marrow erythroid cell smears and professionally annotated to one of the four types of erythroid cells. To distinguish the erythroid cells, one key indicator is the cell shape which is closely related to the cell growth and maturation. Therefore, we design a novel shape-aware image classification network for fine-grained erythroid cell classification. The shape feature is extracted from the shape mask image and aggregated to the raw image feature with a shape attention module. With the shape-attended image feature, our network achieved superior classification performance (81.12\% top-1 accuracy) on the BMEC dataset comparing to the baseline methods. Ablation studies also demonstrate the effectiveness of incorporating the shape information for the fine-grained cell classification. To further verify the generalizability of our method, we tested our network on two additional public white blood cells (WBC) datasets and the results show our shape-aware method can generally outperform recent state-of-the-art works on classifying the WBC. The code and BMEC dataset can be found on https://github.com/wangye8899/BMEC.