Short Shop Schedules

We consider the open shop, job shop, and ow shop scheduling problems with integral processing times. We give polynomial-time algorithms to determine if an instance has a schedule of length at most 3, and show that deciding if there is a schedule of length at most 4 is N P-complete. The latter result implies that, unless P = N P, there does not exist a polynomial-time approximation algorithm for any of these problems that constructs a schedule with length guaranteed to be strictly less than 5/4 times the optimal length. This work constitutes the rst nontrivial theoretical evidence that shop scheduling problems are hard to solve even approximately .