Scheduling with Discretely Compressible Processing Times to Minimize Makespan

In the classical scheduling problems, it is always assumed that the processing time of a job is fixed. In this paper, we address the scheduling with discretely compressible processing times, i.e., the possible processing time of a job can only have finitely many possibilities. Of course, choosing to process any job with a compressed processing time incurs a compression cost. We consider the single machine scheduling problem with discretely compressible processing times. The objective is to minimize makespan with the constraint of total compression cost, i.e., the total compression cost is at most a pre-given constant number. Jobs may have different release times. We also consider the identical-parallel-machine version with simultaneous release times. Since they are all NP-hard, we design for the first time pseudo-polynomial time algorithms and FPTASs. For the first problem, our main approach is dynamic programming and scaling-and-rounding; and for the second one, our approach is dynamic programming and geometry partitioning.