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We propose UTSP, an Unsupervised Learning (UL) framework for solving the Travelling Salesman Problem (TSP). We train a Graph Neural Network (GNN) using a surrogate loss. The GNN outputs a heat map representing the probability for each edge to be part of the optimal path. We then apply local search to generate our final prediction based on the heat map. Our loss function consists of two parts: one pushes the model to find the shortest path and the other serves as a surrogate for the constraint that the route should form a Hamiltonian Cycle. Experimental results show that UTSP outperforms the existing data-driven TSP heuristics.Our approach is parameter efficient as well as data efficient: the model takes $\sim$ 10\% of the number of parameters and $\sim$ 0.2\% of training samples compared with Reinforcement Learning or Supervised Learning methods.

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R is the prize of visited node. Most of MDP is similar with TSP including training scheme. The MDP formulation is mostly same as TSP . This section provides implementation details of the seeder for the experiments. The details of setting T in the inference phase (i.e. in experiments) is described in Appendix A.5. A.3 Detailed Implementation of Reviser This section describes the detailed implementation of the reviser for each target problem.

Large Language Models (LLMs) have shown notable potential in code generation for optimization algorithms, unlocking exciting new opportunities. This paper examines how LLMs, rather than creating algorithms from scratch, can improve existing ones without the need for specialized expertise. To explore this potential, we selected 10 baseline optimization algorithms from various domains (metaheuristics, reinforcement learning, deterministic, and exact methods) to solve the classic Travelling Salesman Problem. The results show that our simple methodology often results in LLM-generated algorithm variants that improve over the baseline algorithms in terms of solution quality, reduction in computational time, and simplification of code complexity, all without requiring specialized optimization knowledge or advanced algorithmic implementation skills.

We propose UTSP, an Unsupervised Learning (UL) framework for solving the Travelling Salesman Problem (TSP). We train a Graph Neural Network (GNN) using a surrogate loss. The GNN outputs a heat map representing the probability for each edge to be part of the optimal path. We then apply local search to generate our final prediction based on the heat map. Our loss function consists of two parts: one pushes the model to find the shortest path and the other serves as a surrogate for the constraint that the route should form a Hamiltonian Cycle.

As a practical and crucial supplement to the classic TSP, it is Existing neural methods for the Travelling Salesman Problem (TSP) highly desired in many real-world scenarios, where a single solution mostly aim at finding a single optimal solution. To discover diverse may be insufficient. For example, 1) when the single target route yet high-quality solutions for Multi-Solution TSP (MSTSP), we propose (solution) becomes unavailable due to unexpected circumstances, a novel deep reinforcement learning based neural solver, which MSTSP offers desirable alternatives; 2) while the single target route is primarily featured by an encoder-decoder structured policy. Concretely, may overlook other important metrics like user preferences, MSTSP on the one hand, a Relativization Filter (RF) is designed to allows for personalized choices among a set of high-quality candidate enhance the robustness of the encoder to affine transformations of routes; 3) while the single target route may incur spontaneous the instances, so as to potentially improve the quality of the found and simultaneous pursuit of the same choice, MSTSP can distribute solutions. On the other hand, a Multi-Attentive Adaptive Active users or loads across different routes, potentially mitigating the jam Search (MA3S) is tailored to allow the decoders to strike a balance and enhancing the overall performance.

Large Language Models (LLMs) are deep learning models designed to generate text based on textual input. Although researchers have been developing these models for more complex tasks such as code generation and general reasoning, few efforts have explored how LLMs can be applied to combinatorial problems. In this research, we investigate the potential of LLMs to solve the Travelling Salesman Problem (TSP). Utilizing GPT-3.5 Turbo, we conducted experiments employing various approaches, including zero-shot in-context learning, few-shot in-context learning, and chain-of-thoughts (CoT). Consequently, we fine-tuned GPT-3.5 Turbo to solve a specific problem size and tested it using a set of various instance sizes. The fine-tuned models demonstrated promising performance on problems identical in size to the training instances and generalized well to larger problems. Furthermore, to improve the performance of the fine-tuned model without incurring additional training costs, we adopted a self-ensemble approach to improve the quality of the solutions.

We introduce a new neural architecture to learn the conditional probability of an output sequence with elements that are discrete tokens corresponding to positions in an input sequence. Such problems cannot be trivially addressed by existent approaches such as sequence-to-sequence [1] and Neural Turing Machines [2], because the number of target classes in each step of the output depends on the length of the input, which is variable. Problems such as sorting variable sized sequences, and various combinatorial optimization problems belong to this class. Our model solves the problem of variable size output dictionaries using a recently proposed mechanism of neural attention. It differs from the previous attention attempts in that, instead of using attention to blend hidden units of an encoder to a context vector at each decoder step, it uses attention as a pointer to select a member of the input sequence as the output.