We study the Online Traveling Salesperson Problem (OLTSP) with predictions. In OLTSP, a sequence of initially unknown requests arrive over time at points (locations) of a metric space. The goal is, starting from a particular point of the metric space (the origin), to serve all these requests while minimizing the total time spent. The server moves with unit speed or is "waiting" (zero speed) at some location. We consider two variants: in the open variant, the goal is achieved when the last request is served. In the closed one, the server additionally has to return to the origin. We adopt a prediction model, introduced for OLTSP on the line, in which the predictions correspond to the locations of the requests and extend it to more general metric spaces. We first propose an oracle-based algorithmic framework, inspired by previous work. This framework allows us to design online algorithms for general metric spaces that provide competitive ratio guarantees which, given perfect predictions, beat the best possible classical guarantee (consistency). Moreover, they degrade gracefully along with the increase in error (smoothness), but always within a constant factor of the best known competitive ratio in the classical case (robustness). Having reduced the problem to designing suitable efficient oracles, we describe how to achieve this for general metric spaces as well as specific metric spaces (rings, trees and flowers), the resulting algorithms being tractable in the latter case. The consistency guarantees of our algorithms are tight in almost all cases, and their smoothness guarantees only suffer a linear dependency on the error, which we show is necessary. Finally, we provide robustness guarantees improving previous results.

翻译：我们研究带预测的在线旅行商问题（OLTSP）。在OLTSP中，一系列初始未知的请求随时间在度量空间的点（位置）上依次到达。目标是从度量空间的特定点（原点）出发，服务所有请求的同时最小化总耗时。服务器以单位速度移动，或在某位置"等待"（零速度）。我们考虑两种变体：开放变体中，最后一个请求被服务时即达成目标；闭合变体中，服务器还需返回原点。我们采用为直线型OLTSP引入的预测模型（其中预测对应请求位置），并将其推广至更一般的度量空间。首先，受先前工作启发，我们提出基于预言机的算法框架。该框架允许我们为一般度量空间设计在线算法，其竞争比保证在完美预测下优于经典情况下的最佳可能保证（一致性）；同时，随着误差增加而优雅退化（平滑性），但始终保持在经典情形下已知最佳竞争比的常数因子范围内（鲁棒性）。在将问题简化为设计合适的有效预言机后，我们描述了如何对一般度量空间以及特定度量空间（环、树和花形）实现这一目标，后者得到的算法具有可处理性。我们的算法在几乎所有情况下的一致性保证都是紧的，其平滑性保证仅对误差产生线性依赖，我们证明这是必要的。最后，我们提供的鲁棒性保证改进了先前结果。