348 European Journal of Operational Research 57 (1992) 348-354 North-Holland Theory and Methodology Task scheduling with interprocessor communication delays Philippe Chr6tienne Universit~ Pierre et Marie Curie, Laboratoire de Mdthodologie et Architecture des Systkmes Informatiques, Tour 55-65 B320, Place Jussieu 4, 75252 Paris Cedex 05, France Received July 1989; revised April 1990 Abstract: Distributed memory architectures raise new and complex scheduling problems. In this paper, we first define a basic distributed memory computer scheduling problem issued from an ideal architec- ture. By proving that the corresponding decision problem is NP-complete, we show that unlike shared memory computer scheduling problems, these new problems do not become easy when the processor limitation constraint is removed. Finally, we improve the knowledge of the borderline between the easy and difficult subproblems of the basic one by giving some polynomial special cases. Keywords: Scheduling, complexity, algorithms, interprocessor communication delay Introduction To efficiently execute an application program on the processors of a distributed memory ma- chine, three difficult problems have to be solved. First, the application program must be split into a set of tasks. Then, each task must be assigned a processor (this is the well-known placement prob- lem). Finally, a starting time must be assigned to each task so that all the precedence and resource constraints are satisfied. In this paper we take as input a set of tasks with precedence constraints and interprocessor communication delays and minimize the make- span of the corresponding scheduling problem [1]. In Section 1 we specify the data of the schedul- ing problem (called FDPS) we get when the num- ber of processors is limited. We then consider the case (called IDPS) when there is no limitation on the number of processors. In Section 2, we prove that the IDPS decision problem is NP-complete; this result shows that, unlike shared memory mul- tiprocessor scheduling problems [2,4], a dis- tributed memory computer scheduling problem does not necessarily become an easy problem when there is no resource limitation. In the last section, we present two polynomial special cases of IDPS. 1. The two problems FDPS and IDPS A generic instance f FDPS is specified in terms of the following five parameters (I, U, p, v, m), where I = (1 ..... n} is a finite set of n tasks; G = (I, U) is a directed acyclic graph; Pi, i~I, is the processing time of task i, whichever processor executes it; vi~ , (i, j) ~ U, is the communication time from task i to task j (i.e., the time needed by task i to transfer data to task j if tasks i and j are as- signed distinct processors); 0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved