A Polynomial Time Approximation Scheme for the Multi-vehicle Scheduling Problem on a Path with Release and Handling Times

• Authors: Yoshiyuki Karuno;Hiroshi Nagamochi
• Venue: ISAAC '01 Proceedings of the 12th International Symposium on Algorithms and Computation
• Year: 2001

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 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 processed 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 present a polynomial time approximation scheme {AƐ} to the problem with a fixed m. Our algorithm can be extended to the case where G is a tree so that a polynomial time approximation scheme is obtained if m and the number of leaves in G are fixed.