Approximation algorithms for solving the heterogeneous Chinese postman problem

Abstract

In this paper, we consider the heterogeneous Chinese postman problem (the HCPP), which generalizes the k-Chinese postman problem. Specifically, given a weighted graph $G=(V,E;w;r)$ with length function $w:E\rightarrow R^{+}$ satisfying the triangle inequality, a fixed depot $r\in V$, and k vehicles having k nonuniform speeds $\lambda _{1}, \lambda _{2}, \ldots ,\lambda _{k}$, respectively, it is asked to find k tours in G for these k vehicles, each starting at the same depot r, and collectively traversing each edge in E at least once. The objective is to minimize the maximum completion time of vehicles, where the completion time of a vehicle is its total travel length divided by its speed. The main contribution of our paper is to show the following two results. (1) Given any small constant $\delta >0$, we design a $20.8765(1+\delta )$-approximation algorithm to solve the HCPP, where the running time required is bounded by a polynomial in the input size and $\frac{1}{\delta }$. (2) We present a $(1+\varDelta -1/k)$-approximation algorithm to solve the HCPP in cubic time, where $\varDelta$ is the ratio of the largest vehicle speed to the smallest one.