On variants of a load-balancing problem with unit-load jobs

In this paper, we reconsider an offline load-balancing problem with unit-time jobs that require one unit from a common resource throughout their execution. In the unit-time case, the jobs have to be assigned to time-slots such that a separable convex function of the load of the resource has to be minimized. Variants of this problem have been studied extensively in the literature under different names. We briefly discuss these problems and give a new implementation for one of them with a better worst-case time complexity than any of the known methods. We also consider the more general preemptive problem in which the execution of the jobs can be interrupted and resumed later. Furthermore, we divide the time horizon into disjoint time intervals, and for each interval, a separable convex cost function is given. The jobs have to be scheduled within their feasible intervals preemptively such that the total cost is minimized, where the cost is determined separately for each interval by the corresponding cost function. We show how to solve this problem in polynomial time by a single minimum-cost-flow computation. For the preemptive problem with one cost function only, we propose a proprietary algorithm for finding a feasible solution which is optimal for any convex cost function. We also present some qualitative computational results.