Complexity and Approximation for the Precedence Constrained Scheduling Problem with Large Communication Delays

We investigate the problem of minimizing the makespan for the multiprocessor scheduling problem. We show that there is no hope of finding a ρ-approximation with $\displaystyle \rho < 1+ 1/(c+4)$ (unless ${\cal{P}}={\cal{NP}}$) for the case where all the tasks of the precedence graph have unit execution times, where the multiprocessor is composed of an unrestricted number of machines, and where c denotes the communication delay between two tasks i and j submitted to a precedence constraint and to be processed by two different machines. The problem becomes polynomial whenever the makespan is at the most (c+1). The (c+2) case is still partially opened.