An Asymptotic $\mathcal{O}$ (ln ρ/ ln ln ρ)-Approximation Algorithm for the Scheduling Problem with Duplication on Large Communication Delay Graphs

Abstract

This article is concerned with the problem of scheduling a parallel application depicted by a precedence graph in presence of large communication delays. The target architecture is constituted of a bounded number m of identical processors linked together by an interconnection network. Communication delays represent the time of data transfer between two tasks of the application allocated to different processors. Our objective is to find an allocation of tasks to the processors and an execution order on each machine such that the overall completion time is minimized. We consider the special case of unit execution time for all computation tasks and a uniform communication delay ρ. We present a new approach based on the reduction of the problem to the successive schedulings of “small graphs”, roughly speaking graphs which can be scheduled in time at most ρ + 1 on an unbounded number of processors. Allowing duplication, corresponding to the recomputation of some of the tasks, this technique allows us to derive an asymptotic $\mathcal{O}$(ln ρ/ ln ln ρ)-approximation algorithm for general precedence graph structure.