Traveling Salesperson Problem

Also known as TSP

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3 papers in the last four weeks, down 25% on the four weeks before. 0.0% of all new papers.

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Latest papers 28

Sep 28, 2026cs.AI

Just Initialize: A Training-Free Initialization Component for Large-Scale Routing Optimization

Large-scale routing problems are difficult to solve efficiently as their search spaces grow rapidly with problem size. Existing approaches primarily improve the optimization procedure itself, often at increasing computational cost. We instead shift the focus to a useful initialization that can be refined into a high-quality solution with limited downstream refinement. We propose Just Initialize, a training-free and solver-agnostic initialization component for large-scale routing optimization. Just Initialize compresses a large routing instance into a compact surrogate space, optimizes its global routing structure, and recovers the resulting solution as an optimization-friendly starting point in the original space. Extensive experiments on Traveling Salesman Problems (TSPs), Capacitated Vehicle Routing Problems (CVRPs), Vehicle Routing Problems with Time Windows (VRPTWs), and Prize-Collecting Traveling Salesman Problems (PCTSPs) demonstrate that Just Initialize achieves high-quality solutions comparable to or better than state-of-the-art methods while substantially reducing computational cost across instances ranging from 1K to 100K nodes, including an average speedup of approximately 70

\times

, sub-second runtimes on 10K-node instances, and runtimes within tens of seconds on 100K-node instances.

Jiale Zhao, Sirui Mao, Zimu Chen +5

School of Automation Science and Electrical Engineering, Beihang University, Beijing, China · School of Computer Science and Engineering, Beihang University, Beijing, China · School of Cyber Science and Technology, Beihang University, Beijing, China +1

Traveling Salesperson Problem Combinatorial Optimization Vehicle Routing

Sep 28, 2026cs.LG

Compute Time Scaling with Recursive Models for Combinatorial Optimization

We propose Tiny Recursive Models for Combinatorial Optimization (\ours{}), a general neural method for combinatorial optimization that scales both depth (how often we recursively invoke our network) and width (how much we sample in parallel). Both are fundamental for combinatorial optimization: hard instances demand a large amount of compute, while a small network is essential to avoid overfitting and capture the algorithmic essence of optimization. In particular, our method consists of a graph-aware tiny recursive model that iterates on a latent state with adaptive halting and needs only a lightweight problem-specific decoder. Compared with previous heatmap-based general neural solvers, it achieves a better balance between solution quality and inference speed on both the Traveling Salesman Problem~(TSP) and the Maximum Independent Set~(MIS) problem, and remains competitive with hybrid methods that combine neural components with heuristics specific to each problem. With the same backbone architecture for both tasks, \ours{} outperforms every diffusion-based solver on TSP from 500 to 10,000 cities at a lower inference cost, and on the standard Erdős--Rényi-[700-800] MIS benchmark it surpasses all neural solvers except those that only work well on MIS. We then explore self-relabeling for self-supervised training. We periodically replace the current set of training labels with the model's own better solutions, as an alternative training signal. Self-relabeling can, while forgoing supervision from near-optimal solutions, still result in on-par quality.

Zhengxi Zhang, Paul Swoboda

Traveling Salesperson Problem Inference-Time Scaling Combinatorial Optimization Neural Combinatorial Optimization

Sep 17, 2026cs.LG

COMPASS: Ordered Clustered Routing at 100K Scale

Large-scale routing often requires visiting clusters of nodes in a prescribed order, giving rise to the Ordered Clustered Traveling Salesman Problem (OCTSP). Optimizing each cluster independently seems natural, but misses non-local dependencies. We introduce the COMPASS algorithm for OCTSP, which combines search with learning-accelerated routing by orchestrating parallel sub-solvers. COMPASS has no quality ceiling and its solutions keep improving with compute. It exploits the clustered structure, and can reach exact solutions in time exponential in cluster size rather than instance size. Empirically, COMPASS consistently outperforms alternative methods. Unlike common large-scale routing solvers, COMPASS consumes general distance matrices and is not limited to coordinate inputs. We demonstrate scaling to 100K synthetic nodes and to 28.5K real e-commerce nodes. To our knowledge, the latter is the largest reported routing solution over asymmetric distances, 9x beyond established ATSP benchmarks.

Ido Greenberg, Hugo Linsenmaier, Piotr Sielski +4

NVIDIA

Traveling Salesperson Problem Combinatorial Optimization Neural Combinatorial Optimization

Sep 2, 2026cs.NE

LLM-Driven Joint Evolution of Coupled Heuristics Components for Routing Optimization

Heuristic design for combinatorial optimization remains heavily reliant on expert knowledge, while existing large language model (LLM)-enhanced evolutionary methods typically evolve isolated algorithmic components, even when one determines the search state on which another operates. This paper proposes LLM-driven Heuristic Components Joint Generation (LLM-HCJG), a population-based framework that jointly generates and co-evolves interdependent heuristic components under a shared design blueprint. Applied to guided local search (GLS), LLM-HCJG couples solution initialization with penalty construction and embeds the generated pair into an enhanced online search mechanism. The resulting form is further transferred from the traveling salesman problem (TSP) to the capacitated vehicle routing problem (CVRP). Theoretical analysis establishes the non-separable state-transition effects between the two components and the advantage in generation consistency. Across synthetic instances and 41 public TSPLIB/CVRPLIB benchmarks, LLM-HCJG attains consistently low optimality gaps, including best or tied-best results on 28 of 29 TSPLIB instances and all 12 CVRPLIB instances. Ablation and structural analyses further indicate that these gains are associated with cross-component compatibility and alignment rather than isolated-component recombination. These results support effective cross-instance transfer within the evaluated routing settings under limited-sample, modest-cost training.

Juntao Wei, Yangming Zhou, Zhibin Jiang +1

Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200030, China · Global Institute of Future Technology, Shanghai Jiao Tong University, Shanghai 200240, China

Automated Heuristic Design Evolutionary Optimization Local Search Vehicle Routing Traveling Salesperson Problem

Aug 13, 2026cs.NE

Insights from Multi-tasking the EAX Algorithm for the Travelling Salesperson Problem

Evolutionary multitasking allows several related problems to be solved in a single run of an algorithm. In this paper, we investigate integrating evolutionary multitasking with Edge Assembly Crossover (MT-EAX) to solve the classical Travelling Salesperson Problem (TSP). To fairly compare MT-EAX against standard EAX under strict compute budgets, we evaluate three scaling methods: generation scaling, population scaling, and balanced scaling. Our results show that generationally scaled MT-EAX is highly effective compute-wise in the early stages of the search, saving

60\%

to

90\%

of compute for equal or better solution quality. We observe that instance geometry has a significant impact, with clustered, normally distributed instances securing larger improvements than uniformly distributed ones. However, when scaling by population or utilising explicit solution transfer, the results are negative due to population starvation and incompatible cross-instance parent selection. We demonstrate that the advantage of MT-EAX derives from increased diversity through parallel search in early generations, which can be successfully preserved using a decoupled configuration to often strictly outperform or match standard EAX performance at final convergence.

Liam Wigney, Aneta Neumann, Yew-Soon Ong +1

Optimisation and Logistics · School of Computer Science and Information Technology · College of Computing & Data Science

Traveling Salesperson Problem Genetic Algorithms

Aug 11, 2026cs.AI

Enhanced Filtering Algorithms for the Euclidean Traveling Salesperson Problem and its variants in Constraint Logic Programming

The Traveling Salesperson Problem (TSP) is one of the best-known problems in computer science and arises in many engineering applications, such as smart vehicles and intelligent transportation systems. In the "Euclidean" case, each node is defined by its coordinates in the plane and distances are computed using the Euclidean metric. In the Constraint Programming (CP) literature, the Euclidean TSP is typically addressed by computing the full distance matrix and treating it as a general case; however this approach ignores the geometric information carried by the points' coordinates. In this work, we propose new filtering algorithms, implemented in Constraint Logic Programming (CLP), that exploit such geometric information to achieve stronger constraint propagation than existing approaches. Moreover, we show how this methodology can be extended to other Euclidean variants of the TSP, including the Euclidean Generalized Traveling Salesperson Problem (EGTSP), which is relevant in practical routing and logistics applications. Experimental results demonstrate the computational advantages of the proposed approach.

Alessandro Bertagnon, Marco Gavanelli

Department of Environmental and Prevention Sciences, University of Ferrara, Italy · Department of Engineering, University of Ferrara, Italy

Traveling Salesperson Problem Constraint Satisfaction Constraint Programming

Aug 10, 2026cs.AI

DualCert: A Solver for the Traveling Salesman Problem with Constraint-Coupled Learning

Large traveling salesman problem (TSP) instances require a solver to allocate limited computation while preserving the validity of its outputs. Existing neural--operations-research (OR) hybrids predict guidance without requiring learned transitions to satisfy constraints discovered during search. DualCert introduces \emph{constraint-coupled learning}, in which current degree equations and dynamically separated subtour-elimination constraints (SECs) define each learned transition. At each refinement, the degree equations and selected, strictly satisfied SEC equations, with positive slacks, define an iterate-dependent primal-slack Karush--Kuhn--Tucker (KKT) manifold. Repaired dual variables and violated SEC rows define a local cost field. An exact constrained mirror-descent step maps each finite state to a positive state on the same manifold. Where selected rows and deterministic ties remain fixed, implicit differentiation maps parameter perturbations into the manifold tangent space and reuses the forward constraint operator for the local-cost-field derivative. The terminal edge state allocates computation across Held--Karp ascent, candidate-graph edge tests, and tour construction under a fixed budget. Deterministic verification recomputes original costs and accepts only verified candidate-graph lower bounds and edge decisions. On 1,000 held-out TSP1000 instances, DualCert attains a mean tour-cost gap of

0.0573\%

from Lin--Kernighan--Helsgaun version 3 (LKH-3) reference tours in

9.55

batch-amortized seconds per instance. It returns a verified candidate-graph lower bound for every instance and achieves

81.46\%

edge-decision coverage. The mean gap is

67.1\%

smaller than the reported NeuroLKH mean gap. Thus, optimization constraints govern learning, while deterministic verification preserves output validity.

Yancheng Song, Yongzhi Qi, Wei Qi +1

Department of Industrial Engineering, Tsinghua University, Beijing 100084, China · JD.com, Beijing, China · College of Engineering, UC Berkeley, Berkeley, CA 94720, USA; Faculty of Engineering, Faculty of Business and Economics, University of Hong Kong, Hong Kong, China

Traveling Salesperson Problem Primal-Dual Optimization Constrained Optimization Neural Combinatorial Optimization

Jul 31, 2026cs.LG

Stabilized Best-of- $K$ Training for Neural Combinatorial Optimization

Leader Reward modifies POMO training to emphasize the best trajectory produced by repeated inference. We test a narrow extension: replace its binary leader/non-leader distinction with a stabilized rank signal indexed by a sampling budget

K

. With the POMO architecture, 3,050-epoch schedule, and TSP-100 test set held fixed, the Leader Reward reimplementation obtains

7.7662

under 100-start, 8-augmentation greedy decoding, matching the reported

7.766

at its displayed precision. Under independent sampling, the stabilized

K=8

recipe lowers realized Best-of-8 cost in all three paired training seeds:

7.7944

versus

7.8136

. This observation is estimation-only and decoder-specific: three seeds are below the six-seed testing floor, Leader Reward is better at sampled

K=1

, and it remains slightly better under its original augmented-greedy protocol. We make no unbiased-estimator, universal superiority, or state-of-the-art claim.

Melveena Jolly, Midhun Xavier

1Independent Researcher

Traveling Salesperson Problem Best-of-N Sampling Combinatorial Optimization RL for Combinatorial Optimization Reward Design for RL Neural Combinatorial Optimization

Jul 31, 2026cs.AI

Geometric Self-Supervised Pre-training for Neural Combinatorial Optimization

Neural Combinatorial Optimization (NCO) techniques have emerged as a highly efficient alternative to traditional exact algorithms for solving routing problems such as the Traveling Salesman Problem (TSP). However, the generalization capabilities of these Reinforcement Learning-based models are severely hindered when scaling to high-dimensional instances. This issue has been mitigated in other domains, like computer vision and natural language processing, by adopting a self-supervised pre-training strategy. Nevertheless, its application to routing graphs, which lack complex topological attributes beyond 2D spatial coordinates, remains a challenge. In this paper, we propose a geometric self-supervised pre-training framework specifically designed to capture spatial invariance and global relative distance distributions. By applying isometric transformations, such as rotations and axial reflections, the model learns robust structural representations prior to the policy optimization phase. Empirical results demonstrate that this strategy consistently outperforms models trained from scratch (baselines), achieving a 7.23% improvement in tour length for massive zero-shot extrapolation scenarios (TSP1,000). Furthermore, the proposed model exhibits remarkable computational efficiency, delivering speedups of up to two orders of magnitude over the exact solver Concorde at massive scales. The source code and pre-trained models are publicly available at https://github.com/davidaguadocosano/TSP-GeoPretrain.git.

David Aguado, Daniel Fuertes, Carlos R. del-Blanco +1

Grupo de Tratamiento de Imágenes, Information Processing and Telecomunications Center, ETSI Telecomunicación, Universidad Politécnica de Madrid, 28040, Madrid, Spain

Traveling Salesperson Problem Self-Supervised Learning Self-Supervised Pre-Training Geometric Representation Learning RL for Combinatorial Optimization Neural Combinatorial Optimization

Jul 26, 2026cs.AI

Understanding Human-like Solutions in Combinatorial Optimization via Learning and Search

Humans often find good solutions to combinatorial optimization problems that are computationally hard even for advanced computer algorithms. In the Euclidean traveling salesman problems (TSP), people rapidly produce tours that are near-optimal, despite severe limits on time and computation. What makes a tour human-like, and how might such solutions be learned? Here we address these questions through a large-scale behavioral and computational investigation of human performance in Euclidean TSP. We sampled a broad space of TSP instances, collected human solutions, and compared them with neural policies based on Pointer Networks, which are recurrent neural networks with an attention-based pointing mechanism that define probability distributions over valid tours. We trained these networks under multiple objectives, including reinforcement learning (RL), supervised learning from optimal tours, supervised learning from human tours, and RL fine-tuning after optimal-supervised pretraining. Human tours were not identical to optimal tours, but occupied a near-optimal geometric basin: they shared many structural properties with optimal solutions while preserving systematic human-specific deviations. The best account of human tours was not direct imitation of optimal tours, but a model pretrained on optimal tours, fine-tuned by RL, and decoded through

\text{Best-of-}N

sampling. These findings suggest that human-like solutions may emerge from a combination of structured supervised learning, RL, and test-time search, echoing computational principles underlying many modern artificial intelligence systems.

Haijiang Yan, Jian-Qiao Zhu, Liqiang Huang +1

Department of Psychology, The University of Warwick, Coventry, United Kingdom. · Department of Psychology, The University of Hong Kong, Hong Kong, China. · Department of Psychology, The Chinese University of Hong Kong, Hong Kong, China. +2

Traveling Salesperson Problem Inference-Time Search Combinatorial Optimization Imitation Learning RL for Combinatorial Optimization Neural Combinatorial Optimization

Jul 21, 2026cs.AI

On the Effectiveness of Pretraining for Graph Combinatorial Optimization

This paper introduces a self-supervised pretraining framework for graph combinatorial optimization specifically designed to address the nature of routing problems like the Traveling Salesman Problem. By utilizing graph contrastive learning with geometric augmentations (specifically, rotations and axial reflections) the model is forced to learn invariant structural representations and global relative distance distributions. Results demonstrate that this pretraining strategy outperforms non-pretrained models across various problem scales. Notably, the hybrid strategy (combining rotation and reflection) achieved a 6.57% improvement in tour length for TSP1000, proving that geometric pretraining is an important inductive bias for effectively scaling neural solvers to high-dimensional instances.

David Aguado, Daniel Fuertes, Carlos R. del-Blanco +1

Grupo de Tratamiento de Imágenes, Information Processing and Telecomunications Center, ETSI Telecomunicación, Universidad Politécnica de Madrid, 28040, Madrid, Spain

Traveling Salesperson Problem Self-Supervised Pre-Training Combinatorial Optimization Graph Contrastive Learning Neural Combinatorial Optimization

Jul 21, 2026cs.LG

Graph Neural Network-based Algorithm Selection for the Traveling Salesman Problem: A Systematic Study of Cost and Rank Losses under Distinct Budget Regimes

Automated Algorithm Selection (AS) aims to improve problem-solving performance by selecting, for each problem instance, the most suitable algorithm from a predefined portfolio. This is particularly relevant to the Traveling Salesman Problem (TSP), where solver performance is strongly instance-dependent. We introduce GNNAS-TSP, a Graph Neural Network (GNN)-based AS framework that learns TSP instance representations directly from raw graph data, avoiding manual feature engineering. GNNAS-TSP formulates AS as a joint cost-prediction and ranking task. We evaluate cost-based (mean squared error (MSE), mean absolute error (MAE), and Huber), rank-based (RankNet, ListNet, and LambdaRank), and hybrid learning objectives for a portfolio comprising Chained Lin-Kernighan, Edge Assembly Crossover, Lin-Kernighan-Helsgaun, Multiagent Optimization System, and Concorde. Experiments use fixed computational budgets of 10 and 60 seconds. On the held-out test set, the selected configurations improve on the Single Best Solver (SBS) in normalized solution cost at both budgets. For the 10s budget, AS achieves substantial and statistically significant cost improvement over SBS. Overall, the results suggest that GNNAS-TSP is a useful meta-solving strategy when exploitable variation exists across solver performance.

Zhaoxuan Li, Jiale Yang, Yifei Lu +1

Duke Kunshan University, 8 Duke Avenue, Kunshan, 215316, China

Traveling Salesperson Problem Algorithm Selection Learning to Rank Combinatorial Optimization

Jul 13, 2026cs.AI

Connected by Construction: Learning Tractable Near-Tour Marginals for Traveling Salesman Problems

Learning-based methods for the traveling salesman problem (TSP) are often evaluated through the tours produced after decoding or search, but the learned object itself frequently lives in a surrogate space such as heatmaps, assignments, construction policies, or search-guidance scores. This hides the fundamental question: what Hamiltonian structure has actually been learned before decoding? In this study, we directly answer this question by learning TSP through a structurally meaningful latent object, rather than leaving most of the Hamiltonian structure to the final decoding stage. Based on a connected-by-construction rooted

1

-tree Gibbs family, we propose an end-to-end unsupervised learning pipeline called \emph{C2TSP}. The pipeline learns residual edge perturbations from unbiased TSP cost through implicit differentiation. For structural correction, a smoothed Held--Karp layer restores expected degree balance, while certificate-guided sharpening further pushes the connected distribution toward more tour-like structures. Experiments show that C2TSP yields strong decoding performance while preserving interpretable structural information. Ablations further verify that edge perturbation and certificate-guided sharpening jointly improve both tour cost and tour-like structure.

Ke Sun, Xinyuan Zhang, Xinwu Qian

Department of Civil and Environmental Engineering, Rice University, Houston, TX

Traveling Salesperson Problem Combinatorial Optimization Neural Combinatorial Optimization

Jul 4, 2026cs.DS

TSP with Predictions: Heatmap to Tour with Provable Guarantees

The Traveling Salesperson Problem (TSP) has long served as a benchmark for evaluating the strength of optimization techniques in the classical theory of algorithms. In recent efforts to apply ML to algorithmic problems, TSP has also become a natural testbed for the development of ML-based techniques. A common approach is to train a neural network to output a heatmap estimating the likelihood of each edge to be part of the optimal tour; however, converting such a heatmap into an actual tour remains a non-trivial and often computationally intensive step. In this work, we propose algorithms for transforming heatmaps into tours with theoretical guarantees linking the achieved approximation ratio to the quality of the provided heatmap. In the spirit of algorithms with predictions, our results can be described as

(1+2\fracη{\mathrm{OPT}})

-approximation algorithms, where

η

denotes the L1 distance between the prediction (heatmap) and an optimal solution (tour). Since the previous works lack such explicit guarantees, we compare our approach against them experimentally.

Marek Eliáš, Fabrizio Grandoni, Adam Polak +1

Traveling Salesperson Problem Combinatorial Optimization Learning-Augmented Algorithms Approximation Algorithms Neural Combinatorial Optimization

Jun 23, 2026cs.AI

GES-TSP: Graph Edge Sparsification for TSP

Solving large-scale instances of the Traveling Salesman Problem (TSP) exactly is computationally expensive. Researchers often employ graph sparsification methods to improve computational efficiency. Traditional sparsification methods typically rely on fixed heuristics and fail to fully exploit instance-specific structural information. In this paper, we propose Graph Edge Sparsification (GES), a learning-based sparsification approach for Euclidean TSP. By incorporating geometric structural information and combinatorial optimization technology, our proposed method adaptively generates a sparsification graph for different instances, significantly reducing the graph size and accelerating the solving process. Experimental results demonstrate that our sparsification method can prune up to 95% of edges on the MATILDA dataset, while keeping the solution gap within 1% of the optimal value. Moreover, our approach exhibits strong generalization capability on the TSPLIB benchmark.In some large-scale instances, the pruning rate exceeds 99%, while the optimality gap remains below 1%.

Tianfeng Chen, Xianyue Li

School of Mathematics and Statistics Lanzhou Universtiy

Traveling Salesperson Problem Graph Structure Learning Graph Sparsification Combinatorial Optimization

Jun 22, 2026cs.LG

GeoRouteNet: Geometry-Enhanced Non-Autoregressive Neural Solver for the Traveling Salesman Problem

The traveling salesman problem (TSP) is a canonical NP-hard combinatorial optimization benchmark that tests the representational capacity and generalization of neural solvers. While non-autoregressive (NAR) approaches offer parallel inference, they often lack sufficient geometric inductive bias and stable training signals, leading to degraded performance under cross-scale and cross-distribution shifts. We propose GeoRouteNet, a geometry-enhanced NAR neural solver for Euclidean TSP. On the model side, GeoRouteNet incorporates centered node features, learnable radial distance basis functions, distance-aware graph attention with explicit edge messaging, LayerNorm-SwiGLU feed-forward blocks, and cross-layer attentive residual mixing. On the training side, we design multi-candidate self-comparison reinforcement learning (MCS-RL), which samples multiple candidate tours per instance, constructs adaptive baselines from greedy and peer candidates, and adds winner-candidate guidance with annealed entropy regularization. On 10,000 random TSP50 instances, GeoRouteNet achieves a 0.32% optimality gap under Beam-1000 decoding. On TSP100, the gap is 1.26%. On 27 stratified TSPLIB EUC_2D instances, the overall gap drops from 17.12% (NAR4TSP reproduction) to 3.60%, while batch inference throughput substantially exceeds that of Concorde and LKH3. Ablation studies confirm that geometric structure enhancement and multi-candidate training are complementary: structure improvements dominate cross-distribution gains, while MCS-RL further stabilizes solution quality when paired with a strong geometric encoder.

Xiang Li

College of Computer Science, Yangtze University, Jingzhou, China

Traveling Salesperson Problem Graph Neural Networks Reinforcement Learning Combinatorial Optimization RL for Combinatorial Optimization Neural Combinatorial Optimization

Jun 17, 2026cs.LG

AGDN: Learning to Solve Traveling Salesman Problem with Anisotropic Graph Diffusion Network

The Traveling Salesman Problem (TSP) is a cornerstone of combinatorial optimization and arises in many practical scenarios. Although graph-based learning approaches have been explored for TSP, the question of how to exploit graph structure more effectively remains open. We present the Anisotropic Graph Diffusion Network (AGDN), a new Graph Neural Network framework designed to solve TSP. Our method tackles two central difficulties: (1) the lack of informative topological prior in fully connected TSP graphs, and (2) losing connected nodes in the optimal solution after the commonly used graph sparsification techniques. To overcome these issues, we construct a MixScore transition matrix that merges node similarity with pairwise distance, and we develop an anisotropic graph diffusion strategy that supports efficient information exchange across multiple hops. Comprehensive experiments spanning diverse instance sizes and node distributions show that AGDN consistently outperforms existing methods while keeping computation time competitive. Furthermore, AGDN generalizes well to problem sizes and distributions beyond those seen during training. The implementation is publicly available at: https://github.com/LabRAI/AGDN.

Bolin Shen, Ziwei Huang, Zhiguang Cao +1

Florida State University · Tallahassee, United States · Singapore Management University +1

Traveling Salesperson Problem Graph Neural Networks Combinatorial Optimization Neural Combinatorial Optimization Graph Diffusion

Jun 17, 2026cs.RO

Two-Phase Bilevel Search for the Moving-Target Traveling Salesman Problem with Moving Obstacles

The Moving-Target Traveling Salesman Problem (MT-TSP) seeks a minimum cost trajectory for an agent that departs from a static depot, visits a set of moving targets, each within one of their assigned time windows, and returns to the depot. In this article, we study the Moving-Target Traveling Salesman Problem with Moving Obstacles (MT-TSP-MO), a generalization of the MT-TSP where the agent trajectory must avoid moving obstacles. We present a Mixed-Integer Conic Programming (MICP) formulation that can be solved using off-the-shelf solvers, as well as a fast and scalable Two-Phase Bilevel Search (TPBS) algorithm that computes high-quality feasible solutions for the problem. We evaluate our approaches against an existing baseline algorithm on a broad range of problem instances with up to 40 targets and 40 obstacles. The results demonstrate that both the proposed methods significantly outperform the baseline with respect to success rates, solution costs, and computation time.

Allen George Philip, Anoop Bhat, Sivakumar Rathinam +1

Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123 · Robotics Institute, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, PA 15213, USA · Mechanical Engineering, and Computer Science and Engineering, Texas A&M University, College Station, TX 77843-3123

Traveling Salesperson Problem Combinatorial Optimization Constrained Optimization Dynamic Obstacle Avoidance Robot Motion Planning

Jun 9, 2026cs.LG

Baseline-Free Policy Optimization for Neural Combinatorial Optimization

Neural combinatorial optimization (NCO) trains autoregressive policies to solve routing problems. The standard training algorithm, REINFORCE with a rollout baseline, requires maintaining and periodically updating a frozen copy of the policy for variance reduction. This baseline introduces a structural vulnerability: on harder instances, a poor baseline produces noisy gradient estimates that can destabilize training. We evaluate Group Relative Policy Optimization (GRPO), an algorithm from large language model alignment that eliminates the baseline entirely by normalizing advantages within groups of sampled trajectories. In a controlled comparison of five RL algorithms on TSP and CVRP benchmarks within the RL4CO framework, we find that: (i) GRPO avoids the training collapse observed with REINFORCE on TSP-100, where performance degrades from cost 9.8 to 52.1 immediately after the warmup phase and does not recover under extended training; (ii) at matched gradient updates, GRPO achieves solution quality within 2% of POMO, a strong AM-based multi-start baseline, while requiring no external baseline; and (iii) P3O, a pairwise preference algorithm also from the alignment literature, is competitive on TSP but shows higher variability on CVRP. These results identify GRPO as a promising baseline-free alternative for NCO, particularly in settings where baseline-dependent training becomes fragile.

Carlos S. Sepúlveda, Gonzalo A. Ruz

Traveling Salesperson Problem Group Relative Policy Optimization Combinatorial Optimization Vehicle Routing RL for Combinatorial Optimization Neural Combinatorial Optimization

Jun 8, 2026cs.NE

Hybrid Metaheuristic Combining the Dragonfly Algorithm and Tabu Search for the Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is a classical NP-hard combinatorial optimization problem that aims to find the shortest Hamiltonian cycle visiting each city exactly once and returning to the starting point. This paper proposes a hybrid metaheuristic for the TSP by combining the Dragonfly Algorithm (DA), a swarm-intelligence-based global search method, with Tabu Search (TS), a memory-based local search technique. The proposed method follows a High-Level Relay Hybridization (HRH) scheme, in which DA is first used to explore the solution space and generate a promising initial tour, while TS subsequently refines this solution through neighbourhood-based improvement and tabu memory. The hybrid approach is evaluated on standard TSPLIB benchmark instances, including burma14, att48, and ch150, and compared with standalone DA, standalone TS, and several classical metaheuristics such as Genetic Algorithm, Ant Colony Optimization, Particle Swarm Optimization, and Random Search. A systematic grid-search procedure is also conducted to study the influence of the main hyperparameters on solution quality and execution time. The experimental results indicate that the proposed hybrid can improve tour quality compared with the standalone DA and TS on the tested instances, highlighting the benefit of combining global exploration with local exploitation. However, the results also suggest that performance remains sensitive to parameter settings and problem size, motivating further validation on larger benchmarks and stronger TSP-specific baselines.

Ammar Bouketta

Ecole Nationale Superieure d’Informatique (ESI), Algiers, Algeria

Traveling Salesperson Problem Metaheuristic Optimization Combinatorial Optimization

Jun 8, 2026cs.AI

Leveraging Structural Constraints for Diffusion-based Neural TSP Solvers

Neural combinatorial optimization has recently achieved strong results on the Euclidean Traveling Salesman Problem (TSP) using generative models such as diffusion and consistency models. State-ofthe-art approaches like FT2T combine fast consistency-based prediction with gradient-based inference time refinement. However, gradient search often incurs significant computational overhead and may not align with the discrete structure of feasible solutions. We introduce Projected Consistency Inference (PCI), a plug-and-play, retraining-free alternative that replaces gradient refinement with structure-aware projections: PCI decodes valid Hamiltonian tours from the consistency model output and applies a lightweight local search (e.g., 2-opt). PCI achieves an average optimality gap (OG) of 0.17% on TSP with 500 cities, and 0.31% on TSP with 1000 cities, outperforming FT2T best settings (OG 0.22% and 0.36%, respectively) while reducing the inference time up to 30 to 40%. PCI also exhibits lower variance and memory usage, and can surpass classical heuristics such as LKH3 in rapid solution generation. Our results demonstrate that structure-aware inference time operations provide a practical and principled path for neural TSP solvers, complementing training time objectives.

Mickaël Basson, Philippe Preux

CRIStAL, Scool · Université de Lille, France · CNRS, France +2

Traveling Salesperson Problem Inference-Time Optimization Combinatorial Optimization Local Search Neural Combinatorial Optimization Consistency Models

Jun 2, 2026cs.NE

Beyond Static Priors: Dynamic Neural Guidance for Large-Scale Ant Colony Optimization

Neural-guided Ant Colony Optimization (ACO) suffers from a fundamental training-inference misalignment: policies are typically trained to generate static priors (e.g., heatmaps), yet deployed to guide iterative, long-horizon search processes. In this paper, we present DyNACO, a novel framework that achieves dynamic neural guidance by periodically observing the pheromone distribution and the incumbent solution. To make DyNACO tractable at scale, we pair the policy with a perturbation-based ACO backend and a scope-restricted refinement mechanism that jointly ensure efficacy and stable credit assignment. On TSP, DyNACO scales to 100,000-node instances and outperforms neural baselines while often reducing total runtime compared to the unguided solver. We extend DyNACO to CVRP via a capacity-aware backend, consistently improving the unguided baseline with less than 1% neural overhead. We further provide in-depth analysis validating the model's generalization capabilities and elucidating why dynamic guidance outperforms static priors. Our work underscores the necessity of aligning neural training with iterative search dynamics in learning-guided optimization. The code is available at https://github.com/shoraaa/DyNACO.

Dat Thanh Tran, Van Khu Vu, Yining Ma

Center for AI Research VinUniversity Hanoi, Vietnam · College of Engineering and Computer Science VinUniversity Hanoi, Vietnam · Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, MA, USA

Traveling Salesperson Problem Metaheuristic Optimization Combinatorial Optimization Neural Combinatorial Optimization

May 31, 2026cs.LG

MViewRouter: Internalizing Geometric Equivariance via Multi-view Alternating Attention for Combinatorial Routing

Combinatorial routing problems such as the Traveling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) are fundamental NP-hard problems with broad real-world applications. While recent deep reinforcement learning methods have shown promising performance, they typically handle geometric symmetries only through data augmentation, resulting in inconsistent decisions and limited generalization. To address this issue, we propose MViewRouter, a multi-view framework that internalizes geometric equivariance as a structural inductive bias to achieve invariant decision-making across routing problem variants. Our approach introduces a Multi-view Alternating Attention (MAA) mechanism that enables parallel processing over the

D_4

symmetry group, alternating between intra-view relational modeling and inter-view feature alignment. Furthermore, we optimize the policy via Collective Policy Gradient Aggregation (CPGA), leveraging consensus gradients from multiple symmetric views to stabilize training and accelerate convergence. Experiments on TSP and CVRP benchmarks, as well as real-world TSPLIB instances, demonstrate that MViewRouter achieves competitive solution quality and strong zero-shot generalization.

Shiyan Liu, Bohan Tan, Yaoxin Wu +1

Huazhong University of Science and Technology · Eindhoven University of Technology

Traveling Salesperson Problem Multi-View Learning Equivariant Neural Networks Combinatorial Optimization Vehicle Routing Neural Combinatorial Optimization

May 14, 2026math.OC

Scalable Solution of the Stochastic Multi-path Traveling Salesman Problem via Neural Networks

The multi-path Traveling Salesman Problem with stochastic travel costs arises in hybrid vehicle routing applications designed for Smart City and City Logistics, where multiple paths exist between each pair of locations. Travel times along these paths are typically affected by real-time traffic conditions and therefore modeled as stochastic. The objective of the problem is to determine a Hamiltonian tour that minimizes the expected total travel cost under uncertainty. In this work, we adopt a two-stage stochastic programming formulation. In the first stage, a predefined route specifying the sequence of locations to be visited is determined, while taking into consideration a second-stage recourse problem that selects the optimal path from the feasible set of alternative paths for each pair of locations, once real-time traffic conditions are realized. To reduce the computational burden imposed by the large number of scenarios required to capture travel time uncertainty, the innovation of this work is the integration of neural network-based surrogate models to approximate the expected value of the second-stage recourse problem. Different architectures and training strategies for the neural networks are proposed and analyzed, with performance evaluated in terms of computation time, solution quality, and generalization capability. Preliminary findings demonstrate the enhanced scalability and practical applicability of the approach for complex vehicle routing problems under uncertainty.

Xiaochen Chou, Ludovica Di Marco, Enza Messina

Department of Informatics, Systems and Communication, University of Milano-Bicocca, Viale Sarca 336, Milan, 20125, Italy.

Neural Surrogate Modeling Stochastic Optimization Traveling Salesperson Problem Combinatorial Optimization Vehicle Routing

Apr 22, 2026cs.LG

Machine Learning-based Two-Stage Graph Sparsification for the Travelling Salesman Problem

High-performance TSP solvers such as Lin-Kernighan-Helsgaun (LKH) search within a \emph{candidate graph} -- a small subset of edges pre-selected for the solver -- rather than over the complete graph. The two leading sparsification heuristics,

α

-Nearest and POPMUSIC, each fall short of the density-coverage balance:

α

-Nearest is dense with stable recall, while POPMUSIC is sparser but its recall degrades with scale. Their union closes the recall gap while remaining far below the complete graph in density, leaving room for further reduction. Existing learning-based sparsifiers score edges on the complete graph, an approach that is expensive and largely limited to Euclidean instances. We propose a two-stage method that inverts this logic. Stage1 takes the union of $α$ -Nearest and POPMUSIC, achieving near-perfect recall at ${\sim}6N$ edges. Crucially, the union annotates each edge with its \emph{source provenance} -- whether it was endorsed by $α$ -Nearest, POPMUSIC, or both. Stage2 trains a lightweight classifier on these annotated edges and prunes the lowest-scoring ones. Because dual-source edges are almost always optimal, the learning problem reduces to filtering the single-source subset -- a substantially easier task than classifying all

O(N^2)

edges from scratch. Across four distance types, five spatial distributions, and problem sizes from 50 to 500, the pipeline reduces candidate-graph density by

37

-

47\%

while retaining

{\geq}99.69\%

of optimal-tour edges, and matches or exceeds the coverage of recent Euclidean-only neural sparsifiers at lower density at TSP500.

Bo-Cheng Lin, Yi Mei, Mengjie Zhang

Centre for Data Science and Artificial Intelligence, and the School of Engineering and Computer Science, Victoria University of Wellington, Wellington, 6140, New Zealand

Traveling Salesperson Problem Graph Sparsification Combinatorial Optimization

Apr 16, 2026cs.NE

On the Use of Iterative Problem Solving for the Traveling Salesperson Problem with Changing Time Window Constraints

In many real-world settings, problem instances that need to be solved are quite similar, and knowledge from previous optimization runs can potentially be utilized. We explore this for the Traveling Salesperson problem with time windows (TSPTW), which often arises in settings where the travel-time matrix is fixed but time-window constraints change across related tasks. Existing TSPTW studies, however, have not systematically compared solving such task sequences independently with sequential transfer from previously solved tasks. We address this gap using a multi-task benchmark in which each base instance is expanded into five related tasks under two environments: partial time-window expansion and swap-additive time reassignment. We compare a standard from-scratch protocol with an iterative protocol that initializes each task from the best tour of the previous task, using the popular local search approaches LNS, VNS, and LKH-3 under a common penalized-score objective. Our experimental results show that the iterative protocol is consistently superior in the progressive-relaxation setting and generally competitive under swap-additive changes, with improvements increasing on more difficult instances.

Hy Nguyen, Thanh Nguyen Pham, Helen Yuliana Angmalisang +2

Optimisation and Logistics School of Computer Science and Information Technology Adelaide University Adelaide, Australia

Traveling Salesperson Problem Metaheuristic Optimization Combinatorial Optimization

Jan 19, 2026cs.AI

Graph Neural Networks are Heuristics

Graph neural networks are usually treated as auxiliaries for combinatorial optimization: they imitate algorithms, guide search, or supply scores to classical procedures. We show that this auxiliary role is not intrinsic. A GNN can itself be a heuristic. For the Euclidean Travelling Salesman Problem, we train a non-autoregressive GNN with no labels, rewards, sequential decoding, search, or local improvement. A differentiable Hamiltonian-cycle objective is the only supervision. The trained model produces a complete tour in one forward pass, while dropout and snapshots from a single training trajectory provide solution diversity without engineered moves. The heuristic is therefore learned, not programmed. It is also fast: batched inference remains in the millisecond regime on GPUs. Experiments on TSP100, TSP200, and TSP500 show that the model consistently improves over nearest-neighbor greedy baselines. These results identify unsupervised GNNs as a class of fast learned heuristics for combinatorial optimization.

Yimeng Min, Carla P. Gomes

Department of Computer Science Cornell University Ithaca, NY, USA

Traveling Salesperson Problem Graph Neural Networks Combinatorial Optimization Neural Combinatorial Optimization

May 22, 2024cs.LG

Leader Reward for POMO-Based Neural Combinatorial Optimization

Deep neural networks based on reinforcement learning (RL) for solving combinatorial optimization (CO) problems are developing rapidly and have shown a tendency to approach or even outperform traditional solvers. However, existing methods overlook an important distinction: CO problems differ from other traditional problems in that they focus solely on the optimal solution provided by the model within a specific length of time, rather than considering the overall quality of all solutions generated by the model. In this paper, we propose Leader Reward and apply it during two different training phases of the Policy Optimization with Multiple Optima (POMO) model to enhance the model's ability to generate optimal solutions. This approach is applicable to a variety of CO problems, such as the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP), and the Flexible Flow Shop Problem (FFSP), but also works well with other POMO-based models or inference phase's strategies. We demonstrate that Leader Reward greatly improves the quality of the optimal solutions generated by the model. Specifically, we reduce the POMO's gap to the optimum by more than 100 times on TSP100 with almost no additional computational overhead.

Chaoyang Wang, Pengzhi Cheng, Jingze Li +1

Fudan University

Traveling Salesperson Problem Reward Shaping Policy Optimization Combinatorial Optimization RL for Combinatorial Optimization Neural Combinatorial Optimization