For prohibitively large-scale Travelling Salesman Problems (TSPs), existing algorithms face big challenges in terms of both computational efficiency and solution quality. To address this issue, we propose a hierarchical destroy-and-repair (HDR) approach, which attempts to improve an initial solution by applying a series of carefully designed destroy-and-repair operations. A key innovative concept is the hierarchical search framework, which recursively fixes partial edges and compresses the input instance into a small-scale TSP under some equivalence guarantee. This neat search framework is able to deliver highly competitive solutions within a reasonable time. Fair comparisons based on nineteen famous large-scale instances (with 10,000 to 10,000,000 cities) show that HDR is highly competitive against existing state-of-the-art TSP algorithms, in terms of both efficiency and solution quality. Notably, on two large instances with 3,162,278 and 10,000,000 cities, HDR breaks the world records (i.e., best-known results regardless of computation time), which were previously achieved by LKH and its variants, while HDR is completely independent of LKH. Finally, ablation studies are performed to certify the importance and validity of the hierarchical search framework.

翻译：针对规模极其庞大的旅行商问题(TSPs)，现有算法在计算效率和求解质量方面均面临巨大挑战。为解决这一问题，我们提出了一种层次化破坏与修复(HDR)方法，该方法通过应用一系列精心设计的破坏与修复操作来改进初始解。其核心创新概念在于层次化搜索框架，该框架在某种等价保证下递归地固定部分边，并将输入实例压缩为小规模TSP。这种简洁的搜索框架能够在合理时间内提供极具竞争力的解。基于19个著名大规模实例（包含10,000至10,000,000个城市）的公平比较表明，HDR在效率和求解质量方面均与现有最优TSP算法高度竞争。值得注意的是，在包含3,162,278和10,000,000个城市的两个大规模实例上，HDR打破了此前由LKH及其变体实现的世界纪录（即不考虑计算时间的最佳已知结果），而HDR完全独立于LKH。最后，通过消融研究验证了层次化搜索框架的重要性和有效性。