We revisit the a priori TSP (with independent activation) and prove stronger approximation guarantees than were previously known. In the a priori TSP, we are given a metric space $(V,c)$ and an activation probability $p(v)$ for each customer $v\in V$. We ask for a TSP tour $T$ for $V$ that minimizes the expected length after cutting $T$ short by skipping the inactive customers. All known approximation algorithms select a nonempty subset $S$ of the customers and construct a master route solution, consisting of a TSP tour for $S$ and two edges connecting every customer $v\in V\setminus S$ to a nearest customer in $S$. We address the following questions. If we randomly sample the subset $S$, what should be the sampling probabilities? How much worse than the optimum can the best master route solution be? The answers to these questions (we provide almost matching lower and upper bounds) lead to improved approximation guarantees: less than 3.1 with randomized sampling, and less than 5.9 with a deterministic polynomial-time algorithm.

翻译：我们重新审视先验旅行商问题（具有独立激活机制），并证明了比先前已知更强的近似保证。在先验TSP中，给定一个度量空间$(V,c)$以及每个客户$v\in V$的激活概率$p(v)$。我们要求找到$V$的一个TSP回路$T$，该回路通过跳过未激活客户来截断$T$，从而最小化期望长度。所有已知的近似算法都选择一个非空客户子集$S$，并构建一个主路由方案，该方案包含$S$的TSP回路以及两条边，将每个客户$v\in V\setminus S$连接到$S$中最近的客户。我们探讨以下问题：如果随机抽样子集$S$，抽样概率应如何确定？最佳主路由方案与最优解相比可能差多少？对这些问题的解答（我们给出了几乎匹配的上下界）导致了改进的近似保证：随机抽样下小于3.1，确定性多项式时间算法下小于5.9。