The problem presented in this paper is a generalization of the usual coupled-tasks scheduling problem in presence of compatibility constraints. The reason behind this study is the data acquisition problem for a submarine torpedo. We investigate a particular configuration for coupled tasks (any task is divided into two sub-tasks separated by an idle time), in which the idle time of a coupled task is equal to the sum of durations of its two sub-tasks. We prove @math -completeness of the minimization of the schedule length, we show that finding a solution to our problem amounts to solving a graph problem, which in itself is close to the minimum-disjoint-path cover (min-DCP) problem. We design a @math -approximation, where a and b (the processing time of the two sub-tasks) are two input data such as a>b>0, and that leads to a ratio between @math and @math . Using a polynomial-time algorithm developed for some class of graph of min-DCP, we show that the ratio decreases to @math .
A coupled-task is a job consisting of two distinct operations. These operations require processing in a predetermined order and at a specified interval apart. This paper considers the problem of sequencing n coupled-task jobs on a single machine with the objective of minimizing the makespan. By making assumptions about processing times, we obtain many special cases and explore the complexity of each case. NP-hardness proofs, or polynomial algorithms, are given to all except one of these special cases. The practical scenario from which this problem originated is also discussed. Nous presentons dans cet article un nouveau type de probleme d'ordonnancement sur monoprocesseur avec tâches-couplees en presence d'un graphe de compatibilite. Ce type de probleme est motive par l'etude d'une torpille autonome sous-marine. Dans un premier temps, nous presentons et modelisons la problematique generale, puis nous etudions la complexite de ce type de problemes en presence de contrainte de compatibilite. Des resultats de N P-completude sont obtenus et un algorithme de complexite polynomiale est donne pour resoudre deux problemes en particulier. Enfin apres une synthese de ces resultats, nous presentons une vision globale de la complexite de cette classe de probleme. This paper considers a special case of the coupled-tasks scheduling problem on one processor. The general problems were analyzed in depth by Orman ad Potts [1]. In this paper, we cosider that all processing times are equal to 1, the gap has exact length L, we have precedence constraits, compatibility constraits are introduced and the criterion is to minimize the scheduling length. We use this problem to study the problem of data acquisittion and data treatment of a torpedo under the water. We show that this problem is NP-complete and we propose an ρ-approximation algorithm where ρ ≤ (L+6) 6. This paper introduces a scheduling problem with coupled-tasks in presence of a compatibility graph on a single processor. We investigate a specific configuration, in which the coupled-tasks possess an idle time equal to 2. The complexity of these problems will be studied according to the presence or absence of triangles in the compatibility graph. As an extended matching, we propose a polynomial-time algorithm which consists in minimizing the number of non-covered vertices, by covering vertices with edges or paths of length two in the compatibility graph. This type of covering will be denoted by 2-cover technique. According on the compatibility graph type, the 2-cover technique provides a polynomial-time rho-approximation algorithm with rho=13 12 (resp. rho=10 9) in absence (resp. presence) of triangles. The coupled tasks scheduling problem was originally introduced for modeling complex radar devices. It is still used for controlling such devices and applied in similar applications. This paper considers a problem of coupled tasks scheduling on a single processor, under the assumptions that all processing times are equal to 1, the gap has exact integer length L and the precedence constraints are strict. We prove that the general problem, when L is part of the input and the precedence constraints graph is a general graph, is NP-hard in the strong sense. We also show that the special case when L=2 and the precedence constraints graph is an in-tree or an out-tree, can be solved in O(n) time. Coupled tasks are two-operation tasks, where the two operations are separated by a time interval of fixed duration. Coupled task scheduling problems refer then to the scheduling of a set of coupled tasks on a single machine. Applications of these problems, reported in the literature, arise in connection with radar systems, robotic cells, and in manufacturing. Most of the complexity issues for scheduling coupled tasks have been settled. However, the complexity status has been unknown for the identical coupled task problem, where multiple copies of a single coupled task are to be processed. The purpose of the article is to solve this open problem in the cyclic case, for which we prove the polynomial complexity. The coupled task problem is to schedule n jobs on one machine where each job consists of two subtasks with required delay time between them. The objective is to minimize the makespan. This problem was analyzed in depth by Orman and Potts [3]. They investigated the complexity of different cases depending on the lengths a i and b i of the two subtasks and the delay time L i . Open image in new window Open image in new window -hardness proofs or polynomial algorithms were given for all cases except for the one where a i =a, b i =b and L i =L. In this paper we present an exact algorithm for this problem with time complexity O(nr 2L ) where Open image in new window holds. Therefore the algorithm is linear in the number of jobs for fixed L.