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

Self-Supervised Learning (SSL) for Combinatorial Optimization (CO) is an emerging paradigm for solving combinatorial problems using neural networks. In this paper, we address a central challenge of SSL for CO: solving problems with discrete constraints. We design an end-to-end differentiable framework that enables us to solve discrete constrained optimization problems with neural networks. Concretely, we leverage algorithmic techniques from the literature on convex geometry and Carathéodory's theorem to decompose neural network outputs into convex combinations of polytope corners that correspond to feasible sets. This decomposition-based approach enables self-supervised training but also ensures efficient quality-preserving rounding of the neural net output into feasible solutions. Extensive experiments in cardinality-constrained optimization show that our approach can consistently outperform neural baselines. We further provide workedout examples of how our method can be applied beyond cardinality-constrained problems to a diverse set of combinatorial optimization tasks, including finding independent sets in graphs, and solving matroid-constrained problems.

While diffusion models have shown promise for combinatorial optimization (CO), their inference-time scaling cost-efficiency remains relatively underexplored. Existing methods improve solution quality by increasing denoising steps, but the performance often becomes saturated quickly. This paper proposes GenSCO to systematically scale diffusion solvers by an orthogonal dimension of inference-time computation beyond denoising step expansion, i.e., search-driven generation. GenSCO takes generation as a search operator rather than a complete solving process, where each operator cycle combines solution disruption (via local search operators) and diffusion sampling, enabling iterative exploration of the learned solution space. Rather than over-refining current solutions, this paradigm encourages the model to leave local optima and explore a broader area of the solution space, ensuring a more consistent scaling effect.

Combinatorial problems on graphs have attracted extensive efforts from the machine learning community over the past decade. Despite notable progress in this area under the umbrella of ML4CO, a comprehensive categorization, unified reproducibility, and transparent evaluation protocols are still lacking for the emerging and immense pool of neural CO solvers. In this paper, we establish a modular and streamlined framework benchmarking prevalent neural CO methods, dissecting their design choices via a tri-leveled "paradigm-model-learning" taxonomy to better characterize different approaches. Further, we integrate their shared features and respective strengths to form 3 unified solvers representing global prediction (GP), local construction (LC), and adaptive expansion (AE) mannered neural solvers. We also collate a total of 65 datasets for 7 mainstream CO problems (including both edge-oriented tasks: TSP, ATSP, CVRP, as well as node-oriented: MIS, MCl, MVC, MCut) across scales to facilitate more comparable results among literature. Extensive experiments upon our benchmark reveal a fair and exact performance exhibition indicative of the raw contribution of the learning components in each method, rethinking and insisting that pre-and post-inference heuristic tricks are not supposed to compensate for sub-par capability of the data-driven counterparts. Under this unified benchmark, an up-to-date replication of typical ML4CO methods is maintained, hoping to provide convenient reference and insightful guidelines for both engineering development and academic exploration of the ML4CO community in the future.

Continuously extending combinatorial optimization objectives is a powerful technique commonly applied to the optimization of set functions. However, few such methods exist for extending functions on permutations, despite the fact that many combinatorial optimization problems, such as the quadratic assignment problem (QAP) and the traveling salesperson problem (TSP), are inherently optimization over permutations.

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

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 the 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.

Neural Combinatorial Optimization (NCO) has emerged as a promising learningbased 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 Problems (TSPs) and Capacitated Vehicle Routing Problems (CVRPs) with up to 100K nodes from diverse distributions.

Achieving generalization in neural approaches across different scales and distributions remains a significant challenge for routing problems. A key obstacle is that neural networks often fail to learn robust principles for identifying universal patterns and deriving optimal solutions from diverse instances. In this paper, we first uncover Purity Law, a fundamental structural principle for optimal solutions of routing problems, defining that edge prevalence grows exponentially with the sparsity of surrounding vertices. Statistically and theoretically validated across diverse instances, Purity Law reveals a consistent bias toward local sparsity in global optima. Building on this insight, we propose Purity Policy Optimization (PUPO), a novel training paradigm that explicitly aligns characteristics of neural solutions with Purity Law during the solution construction process to enhance generalization. Extensive experiments demonstrate that PUPO can be seamlessly integrated with popular neural solvers, significantly enhancing their generalization performance without incurring additional computational overhead during inference. The code is available at https://github.com/Kejun0627/PUPO.

The Asymmetric Traveling Salesman Problem (ATSP) is one of the most fundamental and notoriously challenging problems in combinatorial optimization. We propose a novel continuous relaxation framework for ATSP that leverages differentiable constraints to encourage acyclic structures and valid permutations.