We consider the problem of scheduling unit-length jobs on identical parallel machines such that the makespan of the resulting schedule is minimized. Precedence constraints impose a partial order on the jobs, and both communication and precedence delays impose relative timing constraints on dependent jobs. The combination of these two types of timing constraints naturally models the instruction scheduling problem that occurs during software compilation for state-of-the-art VLIW (Very Long Instruction Word) processors and multiprocessor parallel machines.

We present the first known polynomial-time algorithm for the case where the precedence constraint graph is a forest of in-trees (or a forest of out-trees), the number of machines m is fixed, and the delays (which are a function of both the job pair and the machines on which they run) are bounded by a constant D.

Our algorithm relies on a new structural theorem for scheduling jobs with arbitrary precedence constraints. Given an instance with many independent dags, the theorem shows how to convert, in linear time, a schedule S for only the largest dags into a complete schedule that is either optimal or has the same makespan as S.