Scheduling malleable tasks with precedence constraints

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Abstract

In this paper we propose an approximation algorithm for scheduling malleable tasks with precedence constraints. Based on an interesting model for malleable tasks with continuous processor allotments by Prasanna and Musicus (1991, 1994, 1996) [23], [24], [25], we define two natural assumptions for malleable tasks: the processing time of any malleable task is non-increasing in the number of processors allotted, and the speedup is concave in the number of processors. We show that under these assumptions the work function of any malleable task is non-decreasing in the number of processors and is convex in the processing time. Furthermore, we propose a two-phase approximation algorithm for the scheduling problem. In the first phase we solve a linear program to obtain a fractional allotment for all tasks. By rounding the fractional solution, each malleable task is assigned a number of processors. In the second phase a variant of the list scheduling algorithm is employed. By choosing appropriate values of the parameters, we show (via a nonlinear program) that the approximation ratio of our algorithm is at most 100/63+100(6469+13)/54813.291919. We also show that our result is asymptotically tight.

Keywords

Approximation algorithm
Scheduling
Malleable tasks

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The preliminary version of this paper was published in Proceedings of the 17th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2005), pp. 86–95. This work was performed in part when the second author was with the University of Kiel and with McMaster University. This research was supported in part by the DFG - Graduiertenkolleg, Effiziente Algorithmen und Mehrskalenmethoden; by the EU Thematic Network APPOL I + II, Approximation and Online Algorithms, IST-1999-14084 and IST-2001-32007; by the EU Research Training Network ARACNE, Approximation and Randomized Algorithms in Communication Networks, HPRN-CT-1999-00112; by the EU Project CRESCCO, Critical Resource Sharing for Cooperation in Complex Systems, IST-2001-33135. The second author was also supported by an MITACS grant of Canada; by the NSERC Discovery Grant DG 5-48923; and by the NSERC Industrial Research and Development Fellowship.