From Chinese Postman to Salesman and Beyond I: Approximating Shortest Tours $δ$-Covering All Points on All Edges

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.