Effective and efficient task planning is essential for mobile robots, especially in applications like warehouse retrieval and environmental monitoring. These tasks often involve selecting one location from each of several target clusters, forming a Generalized Traveling Salesman Problem (GTSP) that remains challenging to solve both accurately and efficiently. To address this, we propose a Multimodal Fused Learning (MMFL) framework that leverages both graph and image-based representations to capture complementary aspects of the problem, and learns a policy capable of generating high-quality task planning schemes in real time. Specifically, we first introduce a coordinate-based image builder that transforms GTSP instances into spatially informative representations. We then design an adaptive resolution scaling strategy to enhance adaptability across different problem scales, and develop a multimodal fusion module with dedicated bottlenecks that enables effective integration of geometric and spatial features. Extensive experiments show that our MMFL approach significantly outperforms state-of-the-art methods across various GTSP instances while maintaining the computational efficiency required for real-time robotic applications. Physical robot tests further validate its practical effectiveness in real-world scenarios.
NP hard optimization problems like the Traveling Salesman Problem (TSP) defy efficient solutions in the worst case, yet real-world instances often exhibit exploitable patterns. We propose a novel patternaware complexity framework that quantifies and leverages structural regularities e.g., clustering, symmetry to reduce effective computational complexity across domains, including financial forecasting and LLM optimization. With rigorous definitions, theorems, and a meta learning driven solver pipeline, we introduce metrics like Pattern Utilization Efficiency (PUE) and achieve up to 79 percent solution quality gains in TSP benchmarks (22 to 2392 cities). Distinct from theoretical NP hardness, our approach offers a unified, practical lens for pattern-driven efficiency.
Neural traveling salesman problem (TSP) solvers face two critical challenges: poor generalization for scalable TSPs and high training costs. To address these challenges, we propose a new Rescaling Graph Convolutional Network (RsGCN). Focusing on the scale-dependent features (i.e., features varied with problem scales) related to nodes and edges that influence the sensitivity of GCNs to the problem scales, a Rescaling Mechanism in RsGCN enhances the generalization capability by (1) rescaling adjacent nodes to construct a subgraph with a uniform number of adjacent nodes for each node across various scales of TSPs, which stabilizes the graph message aggregation; (2) rescaling subgraph edges to adjust the lengths of subgraph edges to the same magnitude, which maintains numerical consistency. In addition, an efficient training strategy with a mixed-scale dataset and bidirectional loss is used in RsGCN. To fully exploit the heatmaps generated by RsGCN, we design an efficient post-search algorithm termed Re2Opt, in which a reconstruction process based on adaptive weight is incorporated to help avoid local optima. Based on a combined architecture of RsGCN and Re2Opt, our solver achieves remarkable generalization and low training cost: with only 3 epochs of training on the mixed-scale dataset containing instances with up to 100 nodes, it can be generalized successfully to 10K-node instances without any fine-tuning. Extensive experiments demonstrate our state-of-the-art performance across uniform distribution instances of 9 different scales from 20 to 10K nodes and 78 real-world instances from TSPLIB, while requiring the fewest learnable parameters and training epochs among neural competitors.
Routing problems are canonical combinatorial optimization tasks with wide-ranging applications in logistics, transportation, and supply chain management. However, solving these problems becomes significantly more challenging when complex constraints are involved. In this paper, we propose LMask, a novel learning framework that utilizes dynamic masking to generate high-quality feasible solutions for constrained routing problems. LMask introduces the LazyMask decoding method, which lazily refines feasibility masks with the backtracking mechanism. In addition, it employs the refinement intensity embedding to encode the search trace into the model, mitigating representation ambiguities induced by backtracking. To further reduce sampling cost, LMask sets a backtracking budget during decoding, while constraint violations are penalized in the loss function during training to counteract infeasibility caused by this budget. We provide theoretical guarantees for the validity and probabilistic optimality of our approach. Extensive experiments on the traveling salesman problem with time windows (TSPTW) and TSP with draft limits (TSPDL) demonstrate that LMask achieves state-of-the-art feasibility rates and solution quality, outperforming existing neural methods.
We introduce DyCO-GNN, a novel unsupervised learning framework for Dynamic Combinatorial Optimization that requires no training data beyond the problem instance itself. DyCO-GNN leverages structural similarities across time-evolving graph snapshots to accelerate optimization while maintaining solution quality. We evaluate DyCO-GNN on dynamic maximum cut, maximum independent set, and the traveling salesman problem across diverse datasets of varying sizes, demonstrating its superior performance under tight and moderate time budgets. DyCO-GNN consistently outperforms the baseline methods, achieving high-quality solutions up to 3-60x faster, highlighting its practical effectiveness in rapidly evolving resource-constrained settings.
Algorithms designed for routing problems typically rely on high-quality candidate edges to guide their search, aiming to reduce the search space and enhance the search efficiency. However, many existing algorithms, like the classical Lin-Kernighan-Helsgaun (LKH) algorithm for the Traveling Salesman Problem (TSP), often use predetermined candidate edges that remain static throughout local searches. This rigidity could cause the algorithm to get trapped in local optima, limiting its potential to find better solutions. To address this issue, we propose expanding the candidate sets to include other promising edges, providing them an opportunity for selection. Specifically, we incorporate multi-armed bandit models to dynamically select the most suitable candidate edges in each iteration, enabling LKH to make smarter choices and lead to improved solutions. Extensive experiments on multiple TSP benchmarks show the excellent performance of our method. Moreover, we employ this bandit-based method to LKH-3, an extension of LKH tailored for solving various TSP variant problems, and our method also significantly enhances LKH-3's performance across typical TSP variants.
Dialogue agents that support human users in solving complex tasks have received much attention recently. Many such tasks are NP-hard optimization problems that require careful collaborative exploration of the solution space. We introduce a novel dialogue game in which the agents collaboratively solve a two-player Traveling Salesman problem, along with an agent that combines LLM prompting with symbolic mechanisms for state tracking and grounding. Our best agent solves 45% of games optimally in self-play. It also demonstrates an ability to collaborate successfully with human users and generalize to unfamiliar graphs.
This paper introduces TuneNSearch, a hybrid transfer learning and local search approach for addressing different variants of vehicle routing problems (VRP). Recently, multi-task learning has gained much attention for solving VRP variants. However, this adaptability often compromises the performance of the models. To address this challenge, we first pre-train a reinforcement learning model on the multi-depot VRP, followed by a short fine-tuning phase to adapt it to different variants. By leveraging the complexity of the multi-depot VRP, the pre-trained model learns richer node representations and gains more transferable knowledge compared to models trained on simpler routing problems, such as the traveling salesman problem. TuneNSearch employs, in the first stage, a Transformer-based architecture, augmented with a residual edge-graph attention network to capture the impact of edge distances and residual connections between layers. This architecture allows for a more precise capture of graph-structured data, improving the encoding of VRP's features. After inference, our model is also coupled with a second stage composed of a local search algorithm, which yields substantial performance gains with minimal computational overhead added. Results show that TuneNSearch outperforms many existing state-of-the-art models trained for each VRP variant, requiring only one-fifth of the training epochs. Our approach demonstrates strong generalization, achieving high performance across different tasks, distributions and problem sizes, thus addressing a long-standing gap in the literature.
Reinforcement Learning (RL) has emerged as a powerful tool for neural combinatorial optimization, enabling models to learn heuristics that solve complex problems without requiring expert knowledge. Despite significant progress, existing RL approaches face challenges such as diminishing reward signals and inefficient exploration in vast combinatorial action spaces, leading to inefficiency. In this paper, we propose Preference Optimization, a novel method that transforms quantitative reward signals into qualitative preference signals via statistical comparison modeling, emphasizing the superiority among sampled solutions. Methodologically, by reparameterizing the reward function in terms of policy and utilizing preference models, we formulate an entropy-regularized RL objective that aligns the policy directly with preferences while avoiding intractable computations. Furthermore, we integrate local search techniques into the fine-tuning rather than post-processing to generate high-quality preference pairs, helping the policy escape local optima. Empirical results on various benchmarks, such as the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP) and the Flexible Flow Shop Problem (FFSP), demonstrate that our method significantly outperforms existing RL algorithms, achieving superior convergence efficiency and solution quality.
Achieving generalization in neural approaches across different scales and distributions remains a significant challenge for the Traveling Salesman Problem~(TSP). 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 (PuLa), a fundamental structural principle for optimal TSP solutions, defining that edge prevalence grows exponentially with the sparsity of surrounding vertices. Statistically validated across diverse instances, PuLa 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 PuLa 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.
This study introduces the Iterative Chainlet Partitioning (ICP) algorithm and its neural acceleration for solving the Traveling Salesman Problem with Drone (TSP-D). The proposed ICP algorithm decomposes a TSP-D solution into smaller segments called chainlets, each optimized individually by a dynamic programming subroutine. The chainlet with the highest improvement is updated and the procedure is repeated until no further improvement is possible. The number of subroutine calls is bounded linearly in problem size for the first iteration and remains constant in subsequent iterations, ensuring algorithmic scalability. Empirical results show that ICP outperforms existing algorithms in both solution quality and computational time. Tested over 1,059 benchmark instances, ICP yields an average improvement of 2.75% in solution quality over the previous state-of-the-art algorithm while reducing computational time by 79.8%. The procedure is deterministic, ensuring reliability without requiring multiple runs. The subroutine is the computational bottleneck in the already efficient ICP algorithm. To reduce the necessity of subroutine calls, we integrate a graph neural network (GNN) to predict incremental improvements. We demonstrate that the resulting Neuro ICP (NICP) achieves substantial acceleration while maintaining solution quality. Compared to ICP, NICP reduces the total computational time by 49.7%, while the objective function value increase is limited to 0.12%. The framework's adaptability to various operational constraints makes it a valuable foundation for developing efficient algorithms for truck-drone synchronized routing problems.
Optimization problems often require domain-specific expertise to design problem-dependent methodologies. Recently, several approaches have gained attention by integrating large language models (LLMs) into genetic algorithms. Building on this trend, we introduce Language Model Particle Swarm Optimization (LMPSO), a novel method that incorporates an LLM into the swarm intelligence framework of Particle Swarm Optimization (PSO). In LMPSO, the velocity of each particle is represented as a prompt that generates the next candidate solution, leveraging the capabilities of an LLM to produce solutions in accordance with the PSO paradigm. This integration enables an LLM-driven search process that adheres to the foundational principles of PSO. The proposed LMPSO approach is evaluated across multiple problem domains, including the Traveling Salesman Problem (TSP), heuristic improvement for TSP, and symbolic regression. These problems are traditionally challenging for standard PSO due to the structured nature of their solutions. Experimental results demonstrate that LMPSO is particularly effective for solving problems where solutions are represented as structured sequences, such as mathematical expressions or programmatic constructs. By incorporating LLMs into the PSO framework, LMPSO establishes a new direction in swarm intelligence research. This method not only broadens the applicability of PSO to previously intractable problems but also showcases the potential of LLMs in addressing complex optimization challenges.
In recent years, formulating various combinatorial optimization problems as Quadratic Unconstrained Binary Optimization (QUBO) has gained significant attention as a promising approach for efficiently obtaining optimal or near-optimal solutions. While QUBO offers a general-purpose framework, existing solvers often struggle with performance variability across different problems. This paper (i) theoretically analyzes Mean Field Annealing (MFA) and its variants--which are representative QUBO solvers, and reveals that their underlying self-consistent equations do not necessarily represent the minimum condition of the Kullback-Leibler divergence between the mean-field approximated distribution and the exact distribution, and (ii) proposes a novel method, the Annealed Mean Field Descent (AMFD), which is designed to address this limitation by directly minimizing the divergence. Through extensive experiments on five benchmark combinatorial optimization problems (Maximum Cut Problem, Maximum Independent Set Problem, Traveling Salesman Problem, Quadratic Assignment Problem, and Graph Coloring Problem), we demonstrate that AMFD exhibits superior performance in many cases and reduced problem dependence compared to state-of-the-art QUBO solvers and Gurobi--a state-of-the-art versatile mathematical optimization solver not limited to QUBO.
The Steiner Traveling Salesman Problem (STSP) is a variant of the classical Traveling Salesman Problem. The STSP involves incorporating steiner nodes, which are extra nodes not originally part of the required visit set but that can be added to the route to enhance the overall solution and minimize the total travel cost. Given the NP-hard nature of the STSP, we propose a quantum approach to address it. Specifically, we employ quantum annealing using D-Wave's hardware to explore its potential for solving this problem. To enhance computational feasibility, we develop a preprocessing method that effectively reduces the network size. Our experimental results demonstrate that this reduction technique significantly decreases the problem complexity, making the Quadratic Unconstrained Binary Optimization formulation, the standard input for quantum annealers, better suited for existing quantum hardware. Furthermore, the results highlight the potential of quantum annealing as a promising and innovative approach for solving the STSP.
Recent research at the intersection of quantum computing and routing problems has been highly prolific. Much of this work focuses on classical problems such as the Traveling Salesman Problem and the Vehicle Routing Problem. The practical applicability of these problems depends on the specific objectives and constraints considered. However, it is undeniable that translating complex real-world requirements into these classical formulations often proves challenging. In this paper, we resort to our previously published quantum-classical technique for addressing real-world-oriented routing problems, known as Quantum for Real Package Delivery (Q4RPD), and elaborate on solving additional realistic problem instances. Accordingly, this paper emphasizes the following characteristics: i) simultaneous pickup and deliveries, ii) time-windows, and iii) mobility restrictions by vehicle type. To illustrate the application of Q4RPD, we have conducted an experimentation comprising seven instances, serving as a demonstration of the newly developed features.
Combinatorial optimization problems often rely on heuristic algorithms to generate efficient solutions. However, the manual design of heuristics is resource-intensive and constrained by the designer's expertise. Recent advances in artificial intelligence, particularly large language models (LLMs), have demonstrated the potential to automate heuristic generation through evolutionary frameworks. Recent works focus only on well-known combinatorial optimization problems like the traveling salesman problem and online bin packing problem when designing constructive heuristics. This study investigates whether LLMs can effectively generate heuristics for niche, not yet broadly researched optimization problems, using the unit-load pre-marshalling problem as an example case. We propose the Contextual Evolution of Heuristics (CEoH) framework, an extension of the Evolution of Heuristics (EoH) framework, which incorporates problem-specific descriptions to enhance in-context learning during heuristic generation. Through computational experiments, we evaluate CEoH and EoH and compare the results. Results indicate that CEoH enables smaller LLMs to generate high-quality heuristics more consistently and even outperform larger models. Larger models demonstrate robust performance with or without contextualized prompts. The generated heuristics exhibit scalability to diverse instance configurations.
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.
We offer a new in-depth investigation of global path planning (GPP) for unmanned ground vehicles, an autonomous mining sampling robot named ROMIE. GPP is essential for ROMIE's optimal performance, which is translated into solving the traveling salesman problem, a complex graph theory challenge that is crucial for determining the most effective route to cover all sampling locations in a mining field. This problem is central to enhancing ROMIE's operational efficiency and competitiveness against human labor by optimizing cost and time. The primary aim of this research is to advance GPP by developing, evaluating, and improving a cost-efficient software and web application. We delve into an extensive comparison and analysis of Google operations research (OR)-Tools optimization algorithms. Our study is driven by the goal of applying and testing the limits of OR-Tools capabilities by integrating Reinforcement Learning techniques for the first time. This enables us to compare these methods with OR-Tools, assessing their computational effectiveness and real-world application efficiency. Our analysis seeks to provide insights into the effectiveness and practical application of each technique. Our findings indicate that Q-Learning stands out as the optimal strategy, demonstrating superior efficiency by deviating only 1.2% on average from the optimal solutions across our datasets.
We study the application of emerging photonic and quantum computing architectures to solving the Traveling Salesman Problem (TSP), a well-known NP-hard optimization problem. We investigate several approaches: Simulated Annealing (SA), Quadratic Unconstrained Binary Optimization (QUBO-Ising) methods implemented on quantum annealers and Optical Coherent Ising Machines, as well as the Quantum Approximate Optimization Algorithm (QAOA) and the Quantum Phase Estimation (QPE) algorithm on gate-based quantum computers. QAOA and QPE were tested on the IBM Quantum platform. The QUBO-Ising method was explored using the D-Wave quantum annealer, which operates on superconducting Josephson junctions, and the Quantum Computing Inc (QCi) Dirac-1 entropy quantum optimization machine. Gate-based quantum computers demonstrated accurate results for small TSP instances in simulation. However, real quantum devices are hindered by noise and limited scalability. Circuit complexity grows with problem size, restricting performance to TSP instances with a maximum of 6 nodes. In contrast, Ising-based architectures show improved scalability for larger problem sizes. SQUID-based Ising machines can handle TSP instances with up to 12 nodes, while entropy computing implemented in hybrid optoelectronic components extend this capability to 18 nodes. Nevertheless, the solutions tend to be suboptimal due to hardware limitations and challenges in achieving ground state convergence as the problem size increases. Despite these limitations, Ising machines demonstrate significant time advantages over classical methods, making them a promising candidate for solving larger-scale TSPs efficiently.
Recent advancements in Neural Combinatorial Optimization (NCO) have shown promise in solving routing problems like the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) without handcrafted designs. Research in this domain has explored two primary categories of methods: iterative and non-iterative. While non-iterative methods struggle to generate near-optimal solutions directly, iterative methods simplify the task by learning local search steps. However, existing iterative methods are often limited by restricted neighborhood searches, leading to suboptimal results. To address this limitation, we propose a novel approach that extends the search to larger neighborhoods by learning a destroy-and-repair strategy. Specifically, we introduce a Destroy-and-Repair framework based on Hyper-Graphs (DRHG). This framework reduces consecutive intact edges to hyper-edges, allowing the model to pay more attention to the destroyed part and decrease the complexity of encoding all nodes. Experiments demonstrate that DRHG achieves stateof-the-art performance on TSP with up to 10,000 nodes and shows strong generalization to real-world TSPLib and CVRPLib problems.