Approximation Algorithms for Subclasses of the Makespan Problem on Unrelated Parallel Machines with Restricted Processing Times

Let there be m parallel machines and n jobs, where a job j can be scheduled on machine i to take pi, j ∈ Z+ time units. The makespan Cmax is the completion time of a machine that finishes last. The goal is to produce a schedule with all n jobs that has minimum makespan. This is the makespan problem on unrelated parallel machines, denoted as R||Cmax. Assume p,q ∈ Z+ are constants, let A(p,q) = {a ∈ Z+ | p≤ a≤ q}. We explore a general NP-hard subclass of R||Cmax when processing times are between p and q inclusive or pi, j = ∞ abbreviated as R|pi, j ∈ A(p,q)∪{∞}|Cmax, where pi, j = ∞ means job j cannot be scheduled to machine i. We give a (q/p)-approximation algorithm for R|pi, j ∈ A(p,q)∪{∞}|Cmax. As a consequence, we obtain a 2-approximation algorithm for the NP-hard subclass R||Cmax with job lengths 1, 2, or pi, j = ∞. In addition, we prove R||Cmax with job lengths 1, 2, and 3 is NP-hard, and present a 3/2-approximation algorithm for this subclass of R||Cmax.