A 2-Approximation Algorithm for the Multi-vehicle Scheduling Problem on a Path with Release and Handling Times

In this paper, we consider a scheduling problem of vehicles on a path. Let G = (V, E) be a path, where V = {v1, v2, . . vn} is its set of n vertices and E = {{vj, vj+1} |j = 1, 2, . . .,n - 1} is its set of edges. There are m identical vehicles (1 ≤ m ≤ n). The travel times w(vj, vj+1) (= w(vj+1, vj)) are associated with edges {vj, vj+1} ∈ E. Each job j which is located at each vertex vj ∈ V has release time rj and handling time hj. Any job must be served by exactly one vehicle. The problem asks to find an optimal schedule of m vehicles that minimizes the maximum completion time of all the jobs. The problem is known to be NP-hard for any fixed m ≥ 2. In this paper, we give an O(mn2) time 2-approximation algorithm to the problem, by using properties of optimal gapless schedules.