The Traveling Salesman Problem (TSP) is one of the most extensively studied NP-hard optimization problems. Genetic Algorithm (GA)-based solvers, such as the Edge Assembly Crossover (EAX), achieve state-of-the-art performance on many benchmark instances. However, the scalability of these approaches in massively parallel architectures remains limited because crossover operations involve irregular memory access patterns, graph traversals, and sequential dependencies. Existing GPU-based TSP solvers primarily exploit population-level parallelism and are limited to relatively small problem sizes. This work presents a fine-grain GPU implementation of the partition phase of the Generalized Partition Crossover (GPX) operator for large-scale TSP instances. The proposed approach reformulates GPX partitioning as a graph-parallel problem using coalesced memory layouts, ghost-node transformations, and connected-component analysis. The im- plementation parallelizes the union of parent tours, the splitting of degree- four vertices, the deletion of common edges, and the identification of recombining components using CUDA. Experimental results on instances ranging from 10,000 to 2 million cities demonstrate substantial acceleration over a naive sequential CPU imple- mentation. The proposed GPU partitioning achieves speedups between 48x and 625x while significantly reducing memory overhead. The re- sults demonstrate that operator-level parallelism can substantially im- prove the scalability of GA-based TSP solvers on modern many-core architectures.
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.
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.
David Aguado, Daniel Fuertes, Carlos R. del-Blanco +1cs.AI
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.
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.
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.
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.