A well-studied continuous model of graphs, introduced by Dearing and Francis [Transportation Science, 1974], considers each edge as a continuous unit-length interval of points. For $\delta \geq 0$, we introduce the problem $\delta$-Tour, where the objective is to find the shortest tour that comes within a distance of $\delta$ of every point on every edge. It can be observed that 0-Tour is essentially equivalent to the Chinese Postman Problem, which is solvable in polynomial time. In contrast, 1/2-Tour is essentially equivalent to the Graphic Traveling Salesman Problem (TSP), which is NP-hard but admits a constant-factor approximation in polynomial time. We investigate $\delta$-Tour for other values of $\delta$, noting that the problem's behavior and the insights required to understand it differ significantly across various $\delta$ regimes. We design polynomial-time approximation algorithms summarized as follows: (1) For every fixed $0 < \delta < 3/2$, the problem $\delta$-Tour admits a constant-factor approximation. (2) For every fixed $\delta \geq 3/2$, the problem admits an $O(\log{n})$-approximation. (3) If $\delta$ is considered to be part of the input, then the problem admits an $O(\log^3{n})$-approximation. This is the first of two articles on the $\delta$-Tour problem. In the second one we complement the approximation algorithms presented here with inapproximability results and related to parameterized complexity.

翻译：Dearing和Francis[Transportation Science, 1974]引入了一个经过深入研究的连续图模型，该模型将每条边视为点的连续单位长度区间。对于$\delta \geq 0$，我们引入了$\delta$-游历问题，其目标是找到一条最短的游历路线，使得该路线到每条边上每个点的距离都不超过$\delta$。可以观察到，0-游历问题本质上等价于中国邮递员问题，该问题可在多项式时间内求解。相反，1/2-游历问题本质上等价于图上的旅行商问题，该问题是NP难的，但在多项式时间内允许常数因子近似。我们研究了其他$\delta$值下的$\delta$-游历问题，注意到该问题的行为以及理解它所需的见解在不同的$\delta$区间内存在显著差异。我们设计了多项式时间近似算法，总结如下：(1) 对于每个固定的$0 < \delta < 3/2$，$\delta$-游历问题允许常数因子近似。(2) 对于每个固定的$\delta \geq 3/2$，该问题允许$O(\log{n})$-近似。(3) 如果$\delta$被视为输入的一部分，则该问题允许$O(\log^3{n})$-近似。这是关于$\delta$-游历问题的两篇文章中的第一篇。在第二篇文章中，我们将用不可近似性结果和参数化复杂性相关结果来补充本文提出的近似算法。