SGE: Semantically-Guided Exploration for Unstructured Environments via Image-Space Waypoint Sampling

This work introduces Semantically-Guided Exploration (SGE), a modular exploration framework for ground vehicles that integrates pixel-level semantic segmentation into sampling-based waypoint selection and receding-horizon route optimization. Unlike conventional geometric exploration methods, SGE evaluates candidate exploration goals directly in the image space using a semantic-aware utility function that accounts for terrain traversability, obstacle proximity, objects of interest, and depth-based exploration reward. Sampled waypoints are projected into 3D and ordered through a real-time Traveling Salesman Problem (TSP) formulation, enabling receding-horizon goal selection. To address real-world navigation uncertainty, the framework introduces mechanisms, including temporary taboo regions to handle navigation failures and a graph-based relocation strategy for efficient backtracking across explored areas. We evaluate SGE in standardized simulation benchmarks against state-of-the-art exploration planners and demonstrate competitive performance in volumetric coverage, while enabling semantic task biasing that cannot be achieved by purely geometric methods. The framework is further validated through real-world experiments using multiple robotic platforms in indoor campus buildings and in limestone and coal mines. Results show consistent performance and adaptability across platforms and domains.

Unified Branch-and-Bound Search for the Steiner Traveling Salesman Problem on Graphs of Convex Sets

We formalize the Steiner Traveling Salesman Problem (Steiner-TSP) on Graphs of Convex Sets (GCS), which seeks a minimum-cost closed trajectory through required convex sets while allowing optional transit vertices and revisits. To explore the resulting infinite solution space, we propose a unified branch-and-bound search over rooted walk prefixes. Additive lower-bound-graph costs bound committed prefixes, while a cut-separated connected-flow relaxation lower-bounds the residual cost of visiting every remaining target and returning to the root. Under a uniform positive-cost assumption, best-first traversal terminates after finitely many expansions on every feasible instance without an initial incumbent, whereas depth-first traversal does so once a finite incumbent is available. For a user-specified factor $ε\geq1$, a global lower bound certifies that either strategy's incumbent cost is at most $ε$ times the global optimum. We further demonstrate joint sensing-mode, visitation-order, and continuous-trajectory selection for a mobile-manipulator inspection task, including action precedences expressed in linear temporal logic over finite traces (LTL$_f$). Both traversal strategies find feasible solutions on all benchmark instances within 30s with mean certified optimality gaps of 28.1% and 29.7%, respectively, whereas two recent baselines succeed on only about half of the instances

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.

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.

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%.

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.

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.

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.