In the Time-Windows TSP (TW-TSP) we are given requests at different locations on a network; each request is endowed with a reward and an interval of time; the goal is to find a tour that visits as much reward as possible during the corresponding time window. For the online version of this problem, where each request is revealed at the start of its time window, no finite competitive ratio can be obtained. We consider a version of the problem where the algorithm is presented with predictions of where and when the online requests will appear, without any knowledge of the quality of this side information. Vehicle routing problems such as the TW-TSP can be very sensitive to errors or changes in the input due to the hard time-window constraints, and it is unclear whether imperfect predictions can be used to obtain a finite competitive ratio. We show that good performance can be achieved by explicitly building slack into the solution. Our main result is an online algorithm that achieves a competitive ratio logarithmic in the diameter of the underlying network, matching the performance of the best offline algorithm to within factors that depend on the quality of the provided predictions. The competitive ratio degrades smoothly as a function of the quality and we show that this dependence is tight within constant factors.

翻译：在时间窗旅行商问题（TW-TSP）中，我们接收到网络不同位置上的请求；每个请求附带一个奖励和一个时间区间；目标是找到一条尽可能在相应时间窗内访问更多奖励的路径。对于该问题的在线版本（每个请求在其时间窗开始时才被揭示），无法获得有限的竞争比。我们考虑该问题的一个变体，其中算法会收到关于在线请求出现地点和时间的预测，但无法获知这些辅助信息的质量。由于严格的时间窗约束，诸如TW-TSP之类的车辆路径问题对输入的错误或变化非常敏感，且尚不清楚不完美的预测是否能用于获得有限的竞争比。我们证明，通过将松弛量显式地融入解决方案中，可以实现良好的性能。我们的主要成果是一个在线算法，其竞争比与底层网络直径呈对数关系，且在依赖于所提供预测质量的因子范围内，匹配最优离线算法的性能。该竞争比随预测质量的变化而平滑退化，并且我们证明这种依赖关系在常数因子范围内是紧的。