Scheduling precedence-constrained jobs on related machines with communication delay

We consider the problem of scheduling precedence-constrained jobs on uniformly-related machines in the presence of an arbitrary, fixed communication delay. Communication delay is the amount of time that must pass between the completion of a job on one machine and the start of any successor of that job on a different machine. We consider a model that allows job duplication, i.e. processing of the same job on multiple machines, which, as we show, can reduce the length of a schedule (i.e., its makespan) by a logarithmic factor. Our main result is an approximation algorithm for makespan with approximation ratio polylogarithmic in the number of machines and the length of the communication delay, assuming the minimum makespan is at least the delay. Our algorithm is based on rounding a linear programming relaxation for the problem, which includes carefully designed constraints capturing the interaction among communication delay, precedence requirements, varying speeds, and job duplication. To derive a schedule from a solution to the linear program, we balance the benefits of duplication in satisfying precedence constraints early against its drawbacks in increasing overall system load. Our result builds on two previous lines of work, one with communication delay but identical machines (Lepere, Rapine 2002), and the other with uniformly-related machines but no communication delay (Chudak, Shmoys 1999). We next show that the integrality gap of our mathematical program is polylogarithmic in the communication delay. Our gap construction employs expander graphs and exploits a property of robust expansion and its generalization to paths of longer length, which may be of independent interest. Finally, we quantify the advantage of duplication in scheduling with communication delay. We show that the best schedule without duplication can have a larger makespan than the optimal with duplication by a logarithmic factor. Nevertheless, we present a polynomial time algorithm to transform any schedule to a schedule without duplication at the cost of an increase in makespan polylogarithmic in the number of jobs and machines. Together with our makespan approximation algorithm for schedules allowing duplication, this also yields a polylogarithmic-approximation algorithm for the setting where duplication is not allowed.