On Non-Approximability of Coarse-Grained Workflow Grid Scheduling

Scheduling a scientific workflow onto a computational grid is considered. A computational grid can be regarded as a heterogeneous parallel machine such that the speed of each processor varies over time. A scientific workflow can be modeled as a DAG of tasks. This paper focuses on a coarse-grained workflow. So, any communication delay between tasks is negligible because computation time of every task is much longer than the corresponding communication delay. Hence, in this paper, a coarse-grained workflow grid scheduling problem (WSP for short) is defined as an extension of the classical precedence constrained scheduling problem over a uniform parallel machine with processor speed fluctuation. The objective of our problem is to minimize the makespan of a schedule. It is known that no approximation algorithm exist if a grid has a very long period with zero spare computing power. However, such situation seems to be unrealistic. This paper gives a proof that, unless P = NP, WSP is not approximable within a factor of 1.5 even if accurate performance prediction is possible, all processors have the same peak speed, and speed of every processor at any time is restricted to either 50% or 100% of the peak speed. Since the quite restricted problem is not approximable, any more general problem such that accurate performance prediction is impossible and/or processor speed fluctuation pattern is not restricted is also not approximable. So, the proof implies that WSP is not approximable within a factor of 1.5 in realistic grid environment unless P = NP.