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A well studied and difficult class of scheduling problems concerns parallel machines and precedence constraints. In order to model more realistic situations, we consider precedence delays, associating with each precedence constraint a certain amount of time which must elapse between the completion and start times of the corresponding jobs. Release dates, among others, may be modeled in this fashion. We provide the first constant-factor approximation algorithms for the makespan and the total weighted completion time objectives in this general class of problems. These algorithms are rather simple and practical forms of list scheduling. Our analysis also unifies and simplifies that of a number of special cases heretofore separately studied, while actually improving some of the former approximation results.

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This paper defines and studies an approximation algorithm for scheduling tasks with small communication delays on parallel processors. In a first step, a schedule for the relaxed problem instance with an unlimited number of processors is generated. Then this solution is used to solve the resource conflicts during the scheduling phase on m processors, with a rather unusual feature: a feasible task may be tactically delayed, even inducing idleness on a processor in order to wait for a more impoi-tant task.The relative worst case perfonnance of this algorithm is analysed with respect to the ratio between communication times and processing times and to the performance of the relaxed solution. It improves significantly the best known peiformance ratio of an algorithm for this problem (7/3 against 3j. 0-7803-2535-4/95 $4.00 0 1995 I E E E 167 AMS subject classification : 68M20. I IntroductionWith the recent development of parallel architecture arised a new class of scheduling problems in which communication delays between different processors are considered. A precedence relation from task i to task j means that j needs data from i before being started. If these two tasks are not assigned to the same processor, a delay must be considered between the completion of i and the beginning of j to transfer the data. The aim is to find a schedule that minimizes the makespan. We restrict our study to problems for which communication delay do only depend on tasks, and for which any two processors may communicate. hloreover, the problems tackled in this paper do not allow task replication.Several theoretical studies are devoted to this kind of problems (c.f. the two surveys [1][14]). From these results, it seems that problem instances for which communication delays are smaller than or equal to processing times (SCT tasks systems) are easier to handle. Unfortunately, Picouleau proved in [lo] that such problems are stili NP-hard, small though the communication delays are, and even on an unlimited number of processors.However, approximation algorithms have been studied for these problems, most of them assuming unit processing times and unit communication delays (UET-UCT task systems). Generalizations of classical list scheduling algorithms [ll, 4, 17, 161, as well as clustering algorithms [lS] have been proposed but they all perform quite as bas as possible in the worst case, especially if a great number of processors is assumed. Indeed, the performance ratio may be around 2 if there are enough processors for all tasks, and tends to be 3 on fixed m processors. Section 2 presents SCT task systems and recalls the performance of usual list scheduling algorithms.This paper generalizes to SCT task systems an approximation algorithm 7. 4 I * with a performance ratio --that has been developed for UET-UCT in 3nz@I. --Section 3 presents an integer linear program that models the scheduling problem on an unlimited number of processors. A solution is then built from