The generalized travelling salesperson problem is an important NP-hard combinatorial optimization problem for which metaheuristics, such as local search and evolutionary algorithms, have been used very successfully. Two hierarchical approaches with different neighbourhood structures, namely a cluster-based approach and a node-based approach, have been proposed by Hu and Raidl (2008) for solving this problem. In this article, local search algorithms and simple evolutionary algorithms based on these approaches are investigated from a theoretical perspective. For local search algorithms, we point out the complementary abilities of the two approaches by presenting instances where they mutually outperform each other. Afterwards, we introduce an instance which is hard for both approaches when initialized on a particular point of the search space, but where a variable neighbourhood search combining them finds the optimal solution in polynomial time. Then we turn our attention to analysing the behaviour of simple evolutionary algorithms that use these approaches. We show that the node-based approach solves the hard instance of the cluster-based approach presented in Corus et al. (2016) in polynomial time. Furthermore, we prove an exponential lower bound on the optimization time of the node-based approach for a class of Euclidean instances.

中文翻译：

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广义旅行商问题的局部搜索理论分析和简单进化算法

广义旅行商问题是一个重要的 NP-hard 组合优化问题，元启发式算法（例如局部搜索和进化算法）已被非常成功地用于解决该问题。Hu and Raidl (2008) 提出了两种具有不同邻域结构的分层方法，即基于集群的方法和基于节点的方法来解决这个问题。在本文中，从理论角度研究了基于这些方法的局部搜索算法和简单进化算法。对于局部搜索算法，我们通过展示它们相互超越的实例来指出这两种方法的互补能力。之后，我们引入了一个实例，当在搜索空间的特定点上初始化时，这两种方法都很难，但是其中组合它们的可变邻域搜索在多项式时间内找到最佳解决方案。然后我们将注意力转向分析使用这些方法的简单进化算法的行为。我们表明，基于节点的方法解决了 Corus 等人提出的基于集群的方法的困难实例。(2016) 在多项式时间内。此外，我们证明了一类欧几里得实例的基于节点的方法的优化时间的指数下界。