We consider the online traveling salesperson problem (TSP), where requests appear online over time on the real line and need to be visited by a server initially located at the origin. We distinguish between closed and open online TSP, depending on whether the server eventually needs to return to the origin or not. While online TSP on the line is a very natural online problem that was introduced more than two decades ago, no tight competitive analysis was known to date. We settle this problem by providing tight bounds on the competitive ratios for both the closed and the open variant of the problem. In particular, for closed online TSP, we provide a 1.64-competitive algorithm, thus matching a known lower bound. For open online TSP, we give a new upper bound as well as a matching lower bound that establish the remarkable competitive ratio of 2.04. Additionally, we consider the online D IAL -A-R IDE problem on the line, where each request needs to be transported to a specified destination. We provide an improved non-preemptive lower bound of 1.75 for this setting, as well as an improved preemptive algorithm with competitive ratio 2.41. Finally, we generalize known and give new complexity results for the underlying offline problems. In particular, we give an algorithm with running time O ( n 2 ) for closed offline TSP on the line with release dates and show that both variants of offline D IAL -A-R IDE on the line are NP-hard for any capacity c ≥ 2 of the server.

中文翻译：

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在线 TSP 的严格界限

我们考虑在线旅行销售员问题（TSP），其中请求随着时间的推移在线出现在真实线路上，并且需要由最初位于起点的服务器访问。我们区分封闭式和开放式在线TSP，取决于服务器最终是否需要回源。虽然在线 TSP 是一个非常自然的在线问题，早在 20 多年前就引入了，但迄今为止还没有严格的竞争分析。我们通过为问题的封闭变体和开放变体提供竞争比率的严格限制来解决这个问题。特别是，对于封闭的在线 TSP，我们提供了一个 1.64 竞争算法，从而匹配一个已知的下限。对于开放在线 TSP，我们给出了一个新的上限和一个匹配的下限，建立了 2.04 的显着竞争比率。IAL-ARIDE在线问题，每个请求都需要传输到指定的目的地。我们为此设置提供了改进的非抢占式下限 1.75，以及竞争比为 2.41 的改进的抢占式算法。最后，我们概括已知并为潜在的离线问题提供新的复杂性结果。特别是，我们给出了一个具有运行时间的算法○(n 2) 用于在线上的封闭离线 TSP，并显示离线 D 的两种变体IAL-ARIDE在线上对于任何容量都是 NP-hardC≥2台服务器。