Polynomial complexity of the cyclic identical coupled task problem

Coupled-tasks were introduced by Shapiro in 1980 for scheduling the operations of a radar [19]. The radar emits a pulse that is transmitted to a target and reflected back to the radar, which receives the pulse. Hence the radar must process two operations per task (emission and reception), and those operations are separated by a fixed duration. The coupled task scheduling problem is then to process tasks on a single machine (the radar) with each task j composed of two operations of lengths aj and bj separated by exactly Lj time units. The objective is to minimize the makespan in the non-cyclic case (given a fixed number of coupled tasks) or the throughput rate in the cyclic case (infinite number of tasks). Coupled-task problems belong to the wider class of scheduling multi-operation tasks, where consecutive operations are separated by a certain time interval. In manufacturing processes, the time that has to elapse between operations (delays, time-lags) are often lower bounded (see for instance [12]). We shall here consider only coupled tasks with fixed separation intervals as they apply to radar systems. There is a vast literature on this subject, treating offline and online cases and proposing various algorithms [19,18,17,15,11,10,9]. Let us now consider offline coupled task problems, where a set of tasks {aj ;Lj ; bj} has to be scheduled on a single processor (the radar) with interleaving the coupled tasks but without overlapping the operations. In particular, the case aj = a, Lj = L, bj = b for all j is called the identical coupled task problem. Recently Brauner et al. [8] have described a new application of this coupled task model. They show that this problem is equivalent to a certain no-wait one-machine robotic cell problem usually studied in cyclic mode. For details on the equivalence, see [8]. In particular, producing a large number of single parts corresponds to the identical part production cycle. For this case, studying cyclic identical coupled tasks scheduling problems becomes a more relevant application. The complexity of minimizing the makespan has been described by Orman and Potts [16]. Even the UET problem, aj = 1, Lj , bj = 1 for all j, is NP-hard and