Approximation Algorithms for Scheduling Malleable Tasks under Precedence Constraints

Abstract

This work presents approximation algorithms for scheduling the tasks of a parallel application that are subject to precedence constraints. The considered tasks are malleable which means that they may be executed on a varying number of processors in parallel. The considered objective criterion is the makespan, i.e., the largest task completion time.

We demonstrate a close relationship between this scheduling problem and one of its subproblems, the allotment problem. By exploiting this relationship, we design a polynomial time approximation algorithm with performance guarantee arbitrarily close to ${{\left( {3 + \sqrt 5 } \right)} \mathord{\left/ {\vphantom {{\left( {3 + \sqrt 5 } \right)} {2 \approx 2.61803}}} \right. \kern-\nulldelimiterspace} {2 \approx 2.61803}}$ for the special case of series parallel precedence constraints and for the special case of precedence constraints of bounded width. These special cases cover the important situation of tree structured precedence constraints. For the general case with arbitrary precedence constraints, we give a polynomial time approximation algorithm with performance guarantee $3 + \sqrt 5 \approx 5.23606$.