An Efficient Diffusion-based Non-Autoregressive Solver for Traveling Salesman Problem

TL;DR

DEITSP introduces a diffusion-based non-autoregressive solver for the Traveling Salesman Problem that maps noise directly to an optimal adjacency heatmap in a single step, while also exploring multiple solutions through an efficient iterative noise strategy. It combines a one-step diffusion model with self-consistency, a dual-modality graph transformer for faster, richer feature extraction, and a tailored noise-scheduling framework to balance exploration and refinement. Empirical results show state-of-the-art solution quality and competitive inference speed across small to large TSP instances and real-world distributions, with strong generalization to unseen problem sizes and TSPLIB benchmarks. The approach offers flexible trade-offs between accuracy and latency and demonstrates practical potential for large-scale, real-world routing problems.

Abstract

Recent advances in neural models have shown considerable promise in solving Traveling Salesman Problems (TSPs) without relying on much hand-crafted engineering. However, while non-autoregressive (NAR) approaches benefit from faster inference through parallelism, they typically deliver solutions of inferior quality compared to autoregressive ones. To enhance the solution quality while maintaining fast inference, we propose DEITSP, a diffusion model with efficient iterations tailored for TSP that operates in a NAR manner. Firstly, we introduce a one-step diffusion model that integrates the controlled discrete noise addition process with self-consistency enhancement, enabling optimal solution prediction through simultaneous denoising of multiple solutions. Secondly, we design a dual-modality graph transformer to bolster the extraction and fusion of features from node and edge modalities, while further accelerating the inference with fewer layers. Thirdly, we develop an efficient iterative strategy that alternates between adding and removing noise to improve exploration compared to previous diffusion methods. Additionally, we devise a scheduling framework to progressively refine the solution space by adjusting noise levels, facilitating a smooth search for optimal solutions. Extensive experiments on real-world and large-scale TSP instances demonstrate that DEITSP performs favorably against existing neural approaches in terms of solution quality, inference latency, and generalization ability. Our code is available at $\href{https://github.com/DEITSP/DEITSP}{https://github.com/DEITSP/DEITSP}$.

Figures (11)

• Figure 1: The proposed diffusion model learns to directly map any state in the noise trajectory to its origin state (i.e. the optimal solution), where $\theta$ denotes the neural network and $g$ represents the TSP instance as the condition.
• Figure 2: The network takes node coordinates $v_i$, noisy adjacency matrix $a_{t_{ij}}$ and time step $t$ as inputs to predicts the heatmaps $\tilde{a}_{0_{ij}}$. Mix refers to Eq. (\ref{['eq:mix1']}) and (\ref{['eq:mix2']}), while FFN and Norm is Feed Forward and Layer Normalization used in Vaswani2017
• Figure 3: DDIM iteratively predicts $a_{t-1}$ to progress the chain of mapping noise into data, discarding previously estimated $\tilde{a_0}$. In contrast, our method aggregates the outputs of the diffusion model at multiple steps and decodes all outputs $\tilde{a}_0^t$ to improve solution quality.
• Figure 4: Comparison of the performance of DEITSP with three types of noise schedules and various iteration steps.
• Figure 5: Comparison of noise step values with different iteration schedules (linear, cosine, and inverse function).