DAG Scheduling in the BSP Model

We study the problem of scheduling an arbitrary computational DAG on a fixed number of processors while minimizing the makespan. While previous works have mostly studied this problem in relatively restricted models, we define and analyze DAG scheduling in the Bulk Synchronous Parallel (BSP) model, which is a well-established parallel computing model that captures the communication cost between processors much more accurately. We provide a detailed taxonomy of simpler scheduling models that can be understood as variants or special cases of BSP, and discuss the properties of the problem and the optimum cost in these models, and how they differ from BSP. This essentially allows us to dissect the different building blocks of the BSP model, and gain insight into how each of these influences the scheduling problem. We then analyze the hardness of DAG scheduling in BSP in detail. We show that the problem is solvable in polynomial time for some very simple classes of DAGs, but it is already NP-hard for in-trees or DAGs of height 2. We also separately study the subproblem of scheduling communication steps, and we show that the NP-hardness of this problem can depend on the problem parameters and the communication rules within the BSP model. Finally, we present and analyze a natural formulation of our scheduling task as an Integer Linear Program.