Approximation results for a min–max location-routing problem

This paper studies a min–max location-routing problem, which aims to determine both the home depots and the tours for a set of vehicles to service all the customers in a given weighted graph, so that the maximum working time of the vehicles is minimized. The min–max objective is motivated by the needs of balancing or fairness in vehicle routing applications. We have proved that unless NP=PNP=P, it is impossible for the problem to have an approximation algorithm that achieves an approximation ratio of less than 4/3. Thus, we have developed the first constant ratio approximation algorithm for the problem. Moreover, we have developed new approximation algorithms for several variants, which improve the existing best approximation ratios in the previous literature.