论文标题

带无人机的TSP:弧形弧线的好处

The TSP with drones: The benefits of retraversing the arcs

论文作者

Morandi, Nicola, Leus, Roel, Matuschke, Jannik, Yaman, Hande

论文摘要

在旅行推销员问题中,无人机(TSP-MD),卡车和多个无人机在最短时间内为客户提供服务。无人机由卡车在客户位置发射和检索,他们的每个航班都不能消耗超过电池允许的能量。文献中的大多数问题设置都限制了可行的卡车路线,即封闭的路径,这些路线永远不会访问节点以上。但是,重新访问节点可能会降低为所有客户提供服务所需的时间。此外,我们观察到TSP-MD的最佳解决方案可能会弯曲弧,即,最佳卡车路线可能多次包含相同的弧线。我们将解决方案提到了弧线 - 重新编排,并通过将卡车路线建模为封闭的步行路线,将它们包括在解决方案空间中。我们描述了所有最佳解决方案都在进行欧几里得实例。在先前的研究中似乎并未研究过弧线反向的必要性,而那些允许节点重新访问的人似乎认为始终存在一个没有弧反性的最佳解决方案。我们证明,在文献中通常满足的某些条件下,这个假设是正确的。但是,如果不满足这些条件,则不包括弧线溶液可能会导致最佳价值的增加。我们确定了先验和后验上限在这种增加中保持的案例。最后,我们证明,除非p = np,否则没有可以在恒定因子内近似公制的TSP-MD近似公制的TSP-MD。当卡车可以访问所有节点时,我们明确地确定(非恒定)近似因素。

In the Traveling Salesman Problem with Drones (TSP-mD), a truck and multiple drones cooperate to serve customers in the minimum amount of time. The drones are launched and retrieved by the truck at customer locations, and each of their flights must not consume more energy than allowed by their batteries. Most problem settings in the literature restrict the feasible truck routes to cycles, i.e., closed paths, which never visit a node more than once. Revisiting a node, however, may lower the time required to serve all the customers. Additionally, we observe that optimal solutions for the TSP-mD may retraverse arcs, i.e., optimal truck routes may contain the same arcs multiple times. We refer to such solutions as arc-retraversing, and include them in our solution space by modeling the truck route as a closed walk. We describe Euclidean instances where all the optimal solutions are arc-retraversing. The necessity of arc retraversals does not seem to have been investigated in previous studies, and those that allow node revisits seem to assume that there always exists an optimal solution without arc retraversals. We prove that under certain conditions, which are commonly met in the literature, this assumption is correct. When these conditions are not met, however, excluding arc-retraversing solutions might result in an increase of the optimal value; we identify cases where a priori and a posteriori upper bounds hold on such increase. Finally, we prove that there is no polynomial-time heuristic that can approximate the metric TSP-mD within a constant factor, unless P=NP. We identify a (non-constant) approximation factor explicitly when the truck can visit all the nodes.

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