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.

Vehicle routing problems (VRPs) are significant combinatorial optimization problems (COPs) 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 cross-problem NCO 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 key node connections. 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 tested VRPs. Our ablation study confirms that each component contributes to its cross-problem learning performance. The source code for CaDA is publicly available at https://github.com/CIAM-Group/CaDA.

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

Deep-reinforcement-learning (DRL) based neural combinatorial optimization (NCO) methods have demonstrated efficiency without relying on the guidance of optimal solutions. As the most mainstream among them, the learning constructive heuristic (LCH) achieves high-quality solutions through a rapid autoregressive solution construction process. However, these LCH-based methods are deficient in convergency, and there is still a performance gap compared to the optimal. Intuitively, learning to regret some steps in the solution construction process is helpful to the training efficiency and network representations. This article proposes a novel regret-based mechanism for an advanced solution construction process. Our method can be applied as a plug-in to any existing LCH-based DRL-NCO method. Experimental results demonstrate the capability of our work to enhance the performance of various NCO models. Results also show that the proposed LCH-Regret outperforms the previous modification methods on several typical combinatorial optimization problems. The code and Supplementary File are available at https://github.com/SunnyR7/LCH-Regret.

The min-max vehicle routing problem (min-max VRP) traverses all given customers by assigning several routes and aims to minimize the length of the longest route. Recently, reinforcement learning (RL)-based sequential planning methods have exhibited advantages in solving efficiency and optimality. However, these methods fail to exploit the problem-specific properties in learning representations, resulting in less effective features for decoding optimal routes. This paper considers the sequential planning process of min-max VRPs as two coupled optimization tasks: customer partition for different routes and customer navigation in each route (i.e., partition and navigation). To effectively process min-max VRP instances, we present a novel attention-based Partition-and-Navigation encoder (P&N Encoder) that learns distinct embeddings for partition and navigation. Furthermore, we utilize an inherent symmetry in decoding routes and develop an effective agent-permutation-symmetric (APS) loss function. Experimental results demonstrate that the proposed Decoupling-Partition-Navigation (DPN) method significantly surpasses existing learning-based methods in both single-depot and multi-depot min-max VRPs. Our code is available at

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

As a research hotspot across logistics, operations research, and artificial intelligence, route planning has become a key technology for intelligent transportation systems. Recently, data-driven machine learning heuristics, including learning construction methods and learning improvement methods, have achieved remarkable success in solving single-objective route planning problems. However, many practical route planning scenarios must simultaneously consider multiple conflict objectives. For example, modern logistics companies often need to simultaneously minimize time budget, transportation cost, and vehicle pollution. Several learning construction methods are proposed for solving classical multi-objective route planning (MORP) problems, yet no learning improvement heuristics have been developed so far, even though they are acknowledged to be more efficient in narrowing the optimality gap. To fill this gap, this paper proposes a learning improvement MORP method, Pareto Improver (PI). PI employs a population-based mechanism to approximate the Pareto front with a single deep reinforcement learning model. The experimental results on various MORP problems show that PI can significantly outperform other state-of-the-art methods.