Theoretical aspects of scheduling coupled-tasks in presence of compatibility graph

This paper presents a generalization of the coupled-task scheduling problem introduced by Shapiro (Shapiro 1980), where considered tasks are subject to incompatibility constraint depicted by an undirected graph. The motivation of this problem comes from data acquisition and processing in a mono-processor torpedo used for underwater exploration. As we add the compatibility graph, we focus on complexity of the problem, and more precisely on the limit between the polynomiality and NP-completeness when some other input parameters are restricted (e.g. the ratio between the durations of the two sub-tasks composing a task) : we adapt the global visualization of the complexity of scheduling problems with coupled-task given by Orman and Potts (Orman and Potts 1997) to our problem, determine new complexity results, and thus propose a new visualization including incompatibility constraint. In the end, we give a new polynomialtime approximation algorithm result which completes previous works. Introduction Motivation This paper deals with the problem of data acquisition subject to incompatibility constraint in a submarine torpedo. Many scheduling issues arise in several situations, e.g. in a radar pulsing context (Sherali and Smith 2005; Ageev and Baburin 2007), radar system (Orman, Shahani, and Moore 1998; Orman et al. 1996), or particular application (Brauner et al. 2009). In our context, the torpedo is used to execute submarine several topographic surveys, including topological or temperature measurements. Its aim is to collect data and to process them on a mono-processor within a minimum timeframe. The torpedo realize data acquisition thank to a collection of sensors located on it. Each data acquisition consists in an acquisition task which is divided into two sub-tasks : a sensor first emits a wave which propagates in the water, then he gets a corresponding echo. Scheduling issues appear when several sensors using different frequencies can work in parallel, while acquisitions using the same frequency have to be delayed in order to avoid interferences. It is necessary for robotic engineers to have a good theoretical knowledge of this type of problem. Thus, the aim of the work is to study Copyright c © 2010, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. in many sub-configurations and determine complexity and approximation results on then. Modelisation and related work Coupled-taks (Shapiro 1980) are a natural way to model such data acquisition by our torpedo. Each acquisition task can be viewed as a coupled-task Ai composed by two subtasks, respectively dedicated for wave transmission and echo reception.We note ai and bi the processing time of each subtask. Between these two sub-tasks there is an incompressible and inextensible idle time Li which represents the spread of the echo in the water. due to hardware constraints, we do not work in a preemptive mode : once started, a sub-task cannot be stopped then continued later. A valid schedule implies here that for any task started at t, the first sub-task is fully executed between t and t+ai, and the second between t+ai+Li and t+ai+Li+bi. We note A = {A1, . . . , An} the collection of coupled-tasks to be scheduled. Incompatibility constraint also exists between tasks due to wave interferences. We say two tasks Ai and Aj are compatibles if they use different wave frequencies; thus any sub-task of Ai may be executed during the idle time of Aj , as in Figure 1. We introduce a graphGc = (A, Ec) to model such this compatibility, where edges from Ec link any pair of compatibles coupled-tasks.