Deterministic and randomized scheduling problems under thelp norm on two identical machines

Parallel machine scheduling problems, which are important discrete optimization problems, may occur in many applications. For example, load balancing in network communication channel assignment, parallel processing in large-size computing, task arrangement in flexible manufacturing systems, etc., are multiprocessor scheduling problem. In the traditional parallel machine scheduling problems, it is assumed that the problems are considered in offline or online environment. But in practice, problems are often not really offline or online but somehow in-between. This means that, with respect to the online problem, some further information about the tasks is available, which allows the improvement of the performance of the best possible algorithms. Problems of this class are called semi-online ones. In this paper, the semi-online problemP2|decr|lp (p>1) is considered where jobs come in non-increasing order of their processing times and the objective is to minimize the sum of thelp norm of every machine's load. It is shown thatLS algorithm is optimal for anylp norm, which extends the results known in the literature. Furthermore, randomized lower bounds for the problemsP2|online|lp andP2|decr|lp are presented.