In this paper we study a property of time-dependent graphs, dubbed path ranking invariance. Broadly speaking, a time-dependent graph is path ranking invariant if the ordering of its paths (w.r.t. travel time) is independent of the start time. In this paper we show that, if a graph is path ranking invariant, the solution of a large class of time-dependent vehicle routing problems can be obtained by solving suitably defined (and simpler) time-independent routing problems. We also show how this property can be checked by solving a linear program. If the check fails, the solution of the linear program can be used to determine a tight lower bound. In order to assess the value of these insights, the lower bounds have been embedded into an enumerative scheme. Computational results on the time-dependent versions of the \textit{Travelling Salesman Problem} and the \textit{Rural Postman Problem} show that the new findings allow to outperform state-of-the-art algorithms.

中文翻译：

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关于时间相关图中的路径排名

在本文中，我们研究了时间相关图的一个特性，称为路径排序不变性。从广义上讲，如果其路径的排序（wrt 旅行时间）与开始时间无关，则时间相关图是路径排名不变的。在本文中，我们表明，如果图是路径排序不变的，则可以通过解决适当定义（和更简单）的时间无关路径问题来获得一大类时间相关车辆路径问题的解决方案。我们还展示了如何通过求解线性程序来检查此属性。如果检查失败，则可以使用线性规划的解决方案来确定紧下限。为了评估这些见解的价值，已将下限嵌入到枚举方案中。