Scheduling groups of tasks with precedence constraints on three dedicated processors

Authors:
Renata Mansini;M. Grazia Speranza;Zsolt Tuza
Affiliations:
Dipartimento di Elettronica per l'Automazione, Università di Brescia, via Branze 38, I-25123 Brescia, Italy;Dipartimento Metodi Quantitativi, Università di Brescia, Contrada S. Chiara 48/b, I-25122 Brescia, Italy;Computer and Automation Institute, Hungarian Academy of Sciences, Kende u. 13-17, H-1111 Budapest, Hungary
Venue:
Discrete Applied Mathematics
Year:
2004

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Abstract

We study the problem of scheduling groups of tasks with precedence constraints on three dedicated processors. Each task requires a specified 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 find a nonpreemptive schedule which minimizes the maximum completion time (makespan). This scheduling problem is equivalent to the problem of finding an extension of the constraint graph (i.e. the graph which represents the conflicts 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.