Scheduling groups of tasks with precedence constraints on three dedicated processors

We study the problem of scheduling groups of tasks with precedence constraints on three dedicated processors. Each task requires a speci3ed set of processors. Up to three precedence constraints are considered among groups of tasks requiring the same set of processors. The objective of the problem is to 3nd a nonpreemptive schedule which minimizes the maximum completion time (makespan). This scheduling problem is equivalent to the problem of 3nding an extension of the constraint graph (i.e. the graph which represents the con7icts between tasks and the precedence constraints) to a comparability graph with minimum (over all the extensions) maximum clique weight. The problem is NP-hard in the strong sense. A normal schedule is such that all the tasks requiring the same set of processors are scheduled consecutively. With a normal schedule the problem reduces to the quotient graph of the constraint graph. In this paper we obtain tight approximation results for the minimum makespan of a normal schedule through tight results on the minimum increase of the maximum clique weight when the (partially oriented) quotient graph is extended to a comparability graph. ? 2003 Elsevier B.V. All rights reserved.