A Note on Approximating the Rural Postman Problem below 3/2
Data Structures and Algorithms
Summary
The authors present a new way to solve the rural postman problem, a challenge in finding routes covering specific paths, by improving the best-known approximation solution to be better than 1.5 times the optimal. They do this by adapting a fancy sampling method originally used for the traveling salesman problem. They also note that if you have a good approximation for the traveling salesman problem, you can use it to get a nearly as good solution for the rural postman problem. This connection was hinted at in earlier work but is made clear in their paper.
Authors
Hong Li
Abstract
We give an approximation algorithm for the rural postman problem with approximation ratio strictly smaller than $3/2$. We obtain this result by adapting to the rural postman problem the technique of sampling from maximum entropy distributions for the metric traveling salesman problem of Karlin, Klein, and Oveis Gharan. We also observe that, for every fixed $\varepsilon>0$, any $α$-approximation algorithm for the metric traveling salesman problem yields an $(α+\varepsilon)$-approximation algorithm for the rural postman problem; this implication is already implicit in the treatment of edges that must be traversed in the work of Lampis on the inapproximability of the traveling salesman problem.