Scheduling Unit-time Operation Jobs on Identical Parallel Machines and in a Flow Shop: Complexity and Correlation

This paper considers the complexity and the correlation of scheduling unit-time jobs on identical parallel machines and unit-time ow-shop scheduling under precedence constraints. It is shown that, in the case of chain-like precedence constraints, the nonpre-emptive parallel-machines problem and the ow-shop problem to minimize maximum or total completion time can be solved in polynomial time, and their optimal schedules can be constructed from each other by simple transformations. We show that adding diier-ent release dates of jobs leaves the parallel-machines problem solvable in polynomial time, whereas the related ow-shop problem remains so only in the two-machine case. For general precedence constraints and tree-like precedence constraints with diierent release dates of jobs, it is shown a polynomial-time reduction from the preemptive parallel-machines problems to the ow-shop problems to minimize maximum completion time. The latter are proved to be strongly NP-hard because the former are known to b e so. We also describe related open problems with the maximum completion time and the total completion time criteria.