We present an approximation algorithm for the Prize-collecting Ordered Traveling Salesman Problem (PCOTSP), which simultaneously generalizes the Prize-collecting TSP and the Ordered TSP. The Prize-collecting TSP is well-studied and has a long history, with the current best approximation factor slightly below $1.6$, shown by Blauth, Klein and N\"agele [IPCO 2024]. The best approximation ratio for Ordered TSP is $\frac{3}{2}+\frac{1}{e}$, presented by B\"{o}hm, Friggstad, M\"{o}mke, Spoerhase [SODA 2025] and Armbruster, Mnich, N\"{a}gele [Approx 2024]. The former also present a factor 2.2131 approximation algorithm for Multi-Path-TSP. By carefully tuning the techniques of the latest results on the aforementioned problems and leveraging the unique properties of our problem, we present a 2.097-approximation algorithm for PCOTSP. A key idea in our result is to first sample a set of trees, and then probabilistically pick up some vertices, while using the pruning ideas of Blauth, Klein, N\"{a}gele [IPCO 2024] on other vertices to get cheaper parity correction; the sampling probability and the penalty paid by the LP playing a crucial part in both cases. A straightforward adaptation of the aforementioned pruning ideas would only give minuscule improvements over standard parity correction methods. Instead, we use the specific characteristics of our problem together with properties gained from running a simple combinatorial algorithm to bring the approximation factor below 2.1. Our techniques extend to Prize-collecting Multi-Path TSP, building on results from B\"{o}hm, Friggstad, M\"{o}mke, Spoerhase [SODA 2025], leading to a 2.41-approximation.

翻译：本文针对奖励收集有序旅行商问题提出了一种近似算法，该问题同时推广了奖励收集旅行商问题与有序旅行商问题。奖励收集旅行商问题已有长期研究历史，当前最佳近似因子略低于1.6，由Blauth、Klein和Nägele在IPCO 2024中给出。有序旅行商问题的最佳近似比为3/2+1/e，分别由Böhm、Friggstad、Mömke、Spoerhase在SODA 2025以及Armbruster、Mnich、Nägele在Approx 2024中提出。前者还为多路径旅行商问题提出了2.2131倍近似算法。通过精细调整上述问题最新成果的技术方法，并结合本问题的独特性质，我们为奖励收集有序旅行商问题提出了2.097倍近似算法。我们结果的核心思想是：首先采样一组树结构，然后以概率方式选取部分顶点，同时对其他顶点应用Blauth、Klein、Nägele在IPCO 2024中提出的剪枝思想以获得更经济的奇偶性校正；采样概率与线性规划惩罚项在这两种情况下均起到关键作用。若直接套用前述剪枝思想，仅能较标准奇偶性校正方法获得微小改进。我们通过结合问题特性与运行简单组合算法获得的性质，成功将近似因子降至2.1以下。基于Böhm、Friggstad、Mömke、Spoerhase在SODA 2025中的成果，我们的技术可进一步扩展至奖励收集多路径旅行商问题，获得2.41倍近似算法。