Complexity and Approximation for the Precedence Constrained Scheduling Problem with Large Communication Delays R. Giroudeau, J.C. K¨ onig, F.K. Moula¨ ı, and J. Palaysi LIRMM, 161 rue Ada, 34392 Montpellier Cedex 5, France, UMR 5056 Abstract. We investigate the problem of minimizing the makespan for the multiprocessor scheduling problem. We show that there is no hope of ﬁnding a ρ-approximation with ρ< 1+1/(c + 4) (unless P = NP ) for the case where all the tasks of the precedence graph have unit execution times, where the multiprocessor is composed of an unrestricted number of machines, and where c denotes the communication delay between two tasks i and j submitted to a precedence constraint and to be processed by two diﬀerent machines. The problem becomes polynomial whenever the makespan is at the most (c + 1). The (c + 2) case is still partially opened. 1 Introduction Scheduling theory is concerned with the optimal allocation of scarce resources to activities over time. The theory of the design of algorithms for scheduling is younger, but still has a signiﬁciant history. In this article we adopt the classical scheduling delay model or homogeneous model in which an instance of a scheduling problem is speciﬁed by a set J = {j 1 ,...,j n } of n nonpreemptive tasks, a set of U of q precedence constraints (j i ,j k ) such that G =(J, U ) is a directed acyclic graphs (dag), the processing times p i , ∀j i ∈ J , and the communication times c ik , ∀(j i ,j k ) ∈ U . If the task j i starts its execution at time t on processor π, and if task j k is a successor of j i in the dag, then either j k starts its execution after the time t + p ji on processor π, or after time t + p j k + c jij k on some other processor. In the following we consider the case of ∀j k ∈ J, p j k = 1 and ∀(j i ,j k ) ∈ E, c jij k = c ≥ 2. This model was ﬁrst introduced by Rayward-Smith [13]. In this model we have a set of identical processors that are able to communicate in a uniform way. We want to use these processors in order to process a set of tasks that are subject to precedence constraints. The problem is to ﬁnd a trade-oﬀ between the two extreme solutions, namely, execute all the tasks sequentially without communication, or try to use all the potential parallelism but at the cost of an increased communication overhead. This model has been extensively studies these last years both from the complexity and the (non)-approximability points of view [2]. J.C. Cunha and P.D. Medeiros (Eds.): Euro-Par 2005, LNCS 3648, pp. 252–261, 2005. c  Springer-Verlag Berlin Heidelberg 2005