Scheduling of Independent Dedicated Multiprocessor Tasks ฃ

Algorithms and …, 2002

1997

We study the problem of scheduling a set of n independent multiprocessor tasks with prespecified processor allocations on a fixed number of processors. We propose a linear time algorithm that finds a schedule of minimum makespan in the preemptive model, and a linear time approximation algorithm that finds a schedule of length within a factor of (1+ c) of optimal in the non-preemptive model.

Naval Research Logistics, 1994

In this article we study the problem of scheduling independent tasks, each of which requires the simultaneous availability of a set of prespecified processors, with the objective of minimizing the maximum completion time. We propose a graph-theoretical approach and identify a class of polynomial instances, corresponding to comparability graphs. We show that the scheduling problem is polynomially equivalent to the problem of extending a graph to a comparability graph whose maximum weighted clique has minimum weight. Using this formulation we show that in some cases it is possible to decompose the problem according to the canonical decomposition of the graph. Finally, a general solution procedure is given that includes a branch-and-bound algorithm for the solution of subproblems which can be neither decomposed nor solved in polynomial time. Some examples and computational results are presented. 0 1994 John Wiley & Sons, Inc.

Algorithms

This article extends the scheduling problem with dedicated processors, unit-time tasks, and minimizing maximal lateness Lmax for integer due dates to the scheduling problem, where along with precedence constraints given on the set V={v1,v2, …,vn} of the multiprocessor tasks, a subset of tasks must be processed simultaneously. Contrary to a classical shop-scheduling problem, several processors must fulfill a multiprocessor task. Furthermore, two types of the precedence constraints may be given on the task set V. We prove that the extended scheduling problem with integer release times ri≥0 of the jobs V to minimize schedule length Cmax may be solved as an optimal mixed graph coloring problem that consists of the assignment of a minimal number of colors (positive integers) {1,2, …,t} to the vertices {v1,v2, …,vn}=V of the mixed graph G=(V,A, E) such that, if two vertices vp and vq are joined by the edge [vp,vq]∈E, their colors have to be different. Further, if two vertices vi and vj ar...

Discrete Applied Mathematics, 1994

We investigate the computational complexity of scheduling multiprocessor tasks with prespecified processor allocations. We consider two criteria: minimizing schedule length and minimizing the sum of the task completion times. In addition, we investigate the complexity of problems when precedence constraints or release dates are involved. 1980 Mathematics SUbject Classification (1985 Revision):90835.

IEEE Transactions on Computers, 1986

The problem considered in this paper is the deterministic scheduling of tasks on a set of identical processors. However, the model presented differs from the classical one by the requirement that certain tasks need more than one proc9essor at a time for their processing. This assumption is especially justifidd in some microprocessor applications and its impact on the complexity of minimizing schedule length is studied. First we concentrate on the problem of nonpreemptive scheduling. In this case, polynomial-time algorithms exist only for unit processing times. We present two such algorithms of complexity O(n) for scheduling tasks requiring an arbitrary number of processors between 1 and k at a time where k is a fixed integer. The case for which k is not fixed is shown to be NP-complete. Next, the problem of preemptive scheduling of tasks of arbitrary length is studied. First an algorithm for scheduling tasks requiring one or k processors is presented. Its complexity depends linearly on the number of tasks. Then, the possibility of a linear programming formulation for the general case is analyzed. Index Terms-Complexity analysis, deterministic scheduling, linear programming approach, microprocessor systems, polynomial-in-time algorithms, preemptive and nonpreemptive schedules, schedule length criterion, scheduling multiprocessor tasks. 'We assume that the reader is familiar with the general concepts of the NP-completeness theory which may be found, for example, in [10].

Discrete Optimization, 2005

The paper is concerned with scheduling problems with multiprocessor tasks and presents conditions under which such problems can be solved in polynomial time. The application of these conditions is illustrated by two quite general scheduling problems. These results are complemented by a proof of NP-hardness of the problem with a UET task system, two parallel processors, the criterion of total completion time, and precedence constraints in the form of out-trees.

Information Processing Letters, 1977

2001

Given a set of independent dedicated multiprocessor tasks with time-dependent execution and processor requirements, our objective is to find a schedule minimizing the makespan. In fact, the time-axis is partitionned to a set of intervals, and every task has a specific processing time and processor requirement for every time-interval. We show that this problem (denoted as È Ñ td-fix Ñ Ü) cannot be approximated by any constant ratio approximation algorithm even in the case where two processors are considered (unless È AEÈ). We present a polynomial time approximation scheme (PTAS) for the related-execution-times case with release dates, where the maximum processing time, È Ñ Ü , and the minimum processing time, È Ñ Ò , of each task Ì are such that È Ñ Ü ´ÑµÈ Ñ Ò , with ´Ñµ a parameter depending only on the number of processors Ñ (the number of intervals Ã, as well as Ñ are considered as fixed constants). Notice that this PTAS extends the result of [3] for the classical model with dedicated tasks (È Ñ fix Ñ Ü) in the case where the execution of the tasks is subject to release dates (È Ñ fix Ö Ñ Ü). Furthermore, we show that for the time-dependent problem with dedicated tasks, there is no PTAS in the case where a nonfixed number of intervals is considered even in the relatedexecution-times case and with a fixed number of processors (unless È AEÈ).

Information Processing Letters, 1992