Solving the Traveling Salesperson Problem (TSP) remains a persistent challenge, despite its fundamental role in numerous generalized applications in modern contexts. Heuristic solvers address the demand for finding high-quality solutions efficiently. Among these solvers, the Lin-Kernighan-Helsgaun (LKH) heuristic stands out, as it complements the performance of genetic algorithms across a diverse range of problem instances. However, frequent timeouts on challenging instances hinder the practical applicability of the solver. Within this work, we investigate a previously overlooked factor contributing to many timeouts: The use of a fixed candidate set based on a tree structure. Our investigations reveal that candidate sets based on Hamiltonian circuits contain more optimal edges. We thus propose to integrate this promising initialization strategy, in the form of POPMUSIC, within an efficient restart version of LKH. As confirmed by our experimental studies, this refined TSP heuristic is much more efficient - causing fewer timeouts and improving the performance (in terms of penalized average runtime) by an order of magnitude - and thereby challenges the state of the art in TSP solving.

翻译：求解旅行商问题（TSP）仍然是一个持续的挑战，尽管其在现代众多广义应用中扮演着基础角色。启发式求解器致力于高效地寻找高质量解。在这些求解器中，Lin-Kernighan-Helsgaun（LKH）启发式算法表现突出，因其能在多样化的问题实例上补充遗传算法的性能。然而，在具有挑战性的实例上频繁出现的超时现象阻碍了该求解器的实际应用。在本工作中，我们研究了一个先前被忽视、导致许多超时的因素：基于树结构的固定候选集的使用。我们的研究表明，基于哈密顿回路的候选集包含更多最优边。因此，我们提出将这种有前景的初始化策略——以POPMUSIC的形式——集成到LKH的一个高效重启版本中。正如我们的实验研究所证实，这种改进的TSP启发式算法效率大大提高——导致更少的超时，并将性能（以惩罚平均运行时间计）提升了一个数量级——从而对TSP求解的最新技术水平发起了挑战。