The Traveling Salesman Problem (TSP) is one of the most extensively researched and widely applied combinatorial optimization problems. It is NP-hard even in the symmetric and metric case. Building upon elaborate research, state-of-the-art exact solvers such as CONCORDE can solve TSP instances with several ten thousand vertices. A key ingredient for these integer programming approaches are fast heuristics to find a good initial solution, in particular the Lin-Kernighan-Helsgaun (LKH) heuristic. For instances with few hundred vertices heuristics like LKH often find an optimal solution. In this work we develop variations of LKH that perform significantly better on large instances. LKH repeatedly improves an initially random tour by exchanging edges along alternating circles. Thereby, it respects several criteria designed to quickly find alternating circles that give a feasible improvement of the tour. Among those criteria, the positive gain criterion stayed mostly untouched in previous research. It requires that, while constructing an alternating circle, the total gain has to be positive after each pair of edges. We relax this criterion carefully leading to improvement steps hitherto undiscovered by LKH. We confirm this improvement experimentally via extensive simulations on various benchmark libraries for TSP. Our computational study shows that for large instances our method is on average 13% faster than the latest version of LKH.

翻译：旅行商问题（Traveling Salesman Problem, TSP）是研究最广泛且应用最广泛的组合优化问题之一。即使在对称和度量情形下，该问题仍属于NP-hard。基于深入的研究，当前最先进的精确求解器（如CONCORDE）能够求解包含数万个顶点的TSP实例。这些整数规划方法的关键要素在于快速启发式算法以获取优质初始解，其中尤以Lin-Kernighan-Helsgaun（LKH）启发式最为突出。对于数百个顶点的实例，LKH等启发式算法常能求得最优解。本研究针对大规模实例开发了性能显著提升的LKH变体。LKH通过沿交替环交换边来反复改进初始随机游历，在此过程中遵循若干准则以快速发现能实现可行改进的交替环。在这些准则中，正增益准则（positive gain criterion）在先前研究中几乎保持原样——它要求在构建交替环时，每对边交换后总增益必须保持正值。我们通过谨慎放宽该准则，发现了此前LKH未能探索的改进步骤。通过在多个TSP基准库上的广泛仿真实验，我们验证了这种改进效果。计算研究表明，对于大规模实例，我们的方法平均比最新版LKH快13%。