Online scheduling with rejection and withdrawal

We study an online scheduling problem with rejection, in which some rearrangement of the solution is allowed. This problem is called scheduling with rejection and withdrawal. Each arriving job has a processing time and a rejection cost associated with it, and it needs to be either assigned to a machine or rejected upon arrival. At termination, it is possible to choose at most a fixed number of scheduled jobs and withdraw them (i.e., decide to reject them). We study the minimization version, where the goal is to minimize the sum of the makespan and the total rejection cost (which corresponds to the penalty), and the maximization problem, where the goal is to maximize the sum of the minimum load and the total rejection cost (which corresponds to profit). We study environments of machines, which are the case of m identical machines and the case of two uniformly related machines, and show a strong relation between these problems and the related classic online scheduling problems which they generalize, in contrast to standard scheduling with rejection, which typically makes the scheduling problems harder.