2-Approximation algorithms for the multi-vehicle scheduling problem on a path with release and handling times

In this paper, given a path G with n vertices v1, v2....,vn and m identical vehicles, we consider a scheduling problem of the vehicles on the path. Each vertex vj in G has exactly one job j. Any of the n jobs must be served by some vehicle. Each job j has a release time rj and a handling time hj. A travel time We is associated with each edge e. The problem asks to find an optimal schedule of the m vehicles that minimizes the maximum completion time of all jobs. It is already known that the problem is NP-hard for any fixed m ≥ 2. In this paper, we first present an O(mn2) time 2-approximation algorithm to the problem, by using properties of optimal gapless schedules. We then give a nearly linear time algorithm that delivers a (2 + e)-approximation solution for any fixed e > 0.