Empowering Targeted Neighborhood Search via Hyper Tour for Large-Scale TSP

TL;DR

This paper tackles the memory and guidance bottlenecks of solving large-scale Euclidean TSPs with neural methods. It introduces HyperNS, a framework that combines a sparse heatmap graph, supernode clustering, and a hyper tour as a global abstraction to guide both initialization and iterative optimization, notably reducing search space and memory usage. Key contributions include a hyper-tour guided initialization, a hyper-tour guided targeted neighborhood search, and a memory-efficient architecture that achieves near-linear space complexity while solving up to 71{,}009-city instances with a small gap (about 3.68%), outperforming existing neural methods and offering competitive speed with traditional solvers. The approach demonstrates strong scalability and generalization across synthetic and real-world datasets, with potential applicability to a broader class of combinatorial optimization problems.

Abstract

Traveling Salesman Problem (TSP) is a classic NP-hard problem that has garnered significant attention from both academia and industry. While neural-based methods have shown promise for solving TSPs, they still face challenges in scaling to larger instances, particularly in memory constraints associated with global heatmaps, edge weights, or access matrices, as well as in generating high-quality initial solutions and insufficient global guidance for efficiently navigating vast search spaces. To address these challenges, we propose a Hyper Tour Guided Neighborhood Search (HyperNS) method for large-scale TSP instances. Inspired by the ``clustering first, route second" strategy, our approach initially divides the TSP instance into clusters using a sparse heatmap graph and abstracts them as supernodes, followed by the generation of a hyper tour to guide both the initialization and optimization processes. This method reduces the search space by focusing on edges relevant to the hyper tour, leading to more efficient and effective optimization. Experimental results on both synthetic and real-world datasets demonstrate that our approach outperforms existing neural-based methods, particularly in handling larger-scale instances, offering a significant reduction in the gap to the optimal solution.

Figures (3)

• Figure 1: Overview of our HyperNS framework. Blue edges indicate normal connections, brown edges denote worth-deletion edges that link supernodes. The red edge marks the currently selected worth-deletion edge and its neighboring edges are shown in green. The area enclosed by the red dashed box in the upper-right corner illustrates the targeted neighborhood, which comprises the selected worth-deletion edge and its neighboring edges.
• Figure 2: Illustration of targeted neighborhood search procedure. (a) A TSP tour. (b) Select edge $(A, C)$ along with $A$ and $C$'s $3$-nearest neighbors: $\{T, G, F, B, H, N\}$ and identify all edges involving them as $\mathbb{E}_{del}$. (c) Delete all edges in $\mathbb{E}_{del}$. (d) Retain only vertices $\{U, M, V, D\}$ while adding two new edges, $(U, M)$ and $(V, D)$ to form a reduced subproblem with no more than 12 vertices. (e) Solve the subproblem via LK search, keeping fixed edges $(U, M)$ and $(V, D)$ unchanged. (f) Restore the fixed edges to their previous segments, yielding an improved version of the initial tour.
• Figure 3: Parameter analysis. In each subplot, the left and right y-axes show tour length and total running time, respectively.