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

Abstract

In this paper, we consider a scheduling problem of vehicles on a path. Let G = (V, E) be a path, where V = {v ₁, v ₂,..., v _n } is its set of n vertices and E = {{v _j, v _j+1 | j = 1, 2,..., n-1} is its set of edges. There are m identical vehicles (1 ≤ m ≤ n). The travel times w(v _j, v _j+1) (= w(v _j+1,v _j)) are associated with edges {v _j, v _{j+1} ∈ E}. Each job j which is located at each vertex v _j ∈ V has release time r _j and handling time h _j. 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(mn ²) time 2-approximation algorithm to the problem, by using properties of optimal gapless schedules.