Multiprocessor scheduling with communication delays

Scheduling with Communication Delays

“…S o t h e a i m , i n t h i s m o d e l , i s t o f i n d a c o m p r o m i s e b e t w e e n a s e q u e n t i a l execution and a parallel execution. These two models have been extensively studied over the last few years from both the complexity and the (non)-approximability points of view (see (Graham et al, 1979) and (Chen et al, 1998) (Veltman, 1993). Proof The proof is based on the notion of total unimodularity matrix, see (Veltman, 1993) and see (Schrijver, 1998 (Veltman, 1993).…”

“…with performance bound smaller than 7/6 unless , see (Veltman, 1993). Proof The proof of Corollary 1.2.1 is an immediate consequence of the Impossibility Theorem, (see (Chrétienne and Picouleau, 1995), (Garey and Johnson, 1979)).…”

“…In this section, a lower and upper bound will be presented, Theorem 1.2.4 The problem of deciding whether an instance of , p i = 1, c ij = problem has a schedule of length 3 is polynomial, see (Picouleau, 1995 (Veltman, 1993). Proof The proof is based on the ATP-complete problem Clique.…”

Three, four, five, six, or the complexity of scheduling with communication delays

“…Given a schedule, let Sj and C j denote the starting and completion time of task J j , respectively, for j =1, ... , n. We wish to minimize the length or makespan ofa schedule, that is, the maximum task completion time, defined as C max=maxlSjSnCj ' There are two variants of the problem, depending on whether the number of processors is restrictively small or not. With a modification of the three-field notation scheme that was proposed by Veltman, Lageweg, and Lenstra [1990] as an extension of the terminology introduced by Graham, Lawler, Lenstra, and Rinnooy Kan [1979], we denote the first variant by P Iprec,c = I,pj= II C max and the second variant by poo Iprec,c = I,Pj = II C max ' The P Iprec,c = I,Pj = II C max problem was first addressed by Rayward-Smith [1987], who established NP-hardness and showed that an active schedule is no longer than 3-2/m times the optimum. A schedule is active if no task can start earlier without increasing the start time of another task.…”

Scheduling multiprocessor tasks — An overview

“…To denote the analyzed scheduling problems we will use the standard three-field notation a [ /3 IT proposed in [52] with extensions introduced in [22,76]. Yet, the modifications proposed in [76] are not satisfactory to describe the variety of the considered multiprocessor scheduling problems.…”