Optimizing busy time on parallel machines

We consider the following fundamental parallel machines scheduling problem in which the input consists of n jobs to be scheduled on a set of identical machines of bounded capacity g, which is the maximal number of jobs that can be processed simultaneously by a single machine. Each job is associated with a time interval during which it should be processed from start to end (and in one of our extensions it has to be scheduled also in a continuous number of days; this corresponds to a two-dimensional variant of the problem). We consider two versions of the problem. In the scheduling minimization version the goal is to minimize the total busy time of machines used to schedule all jobs. In the resource allocation maximization version the goal is to maximize the number of jobs that can be scheduled for processing under a budget constraint given in terms of busy time. This is the first study of the maximization version of the problem. The minimization problem is known to be NP-Hard, thus the maximization problem is also NP-Hard. We consider various special cases, identify cases where an optimal solution can be computed in polynomial time, and mainly provide constant factor approximation algorithms for both minimization and maximization problems. Some of our results improve upon the best known results for this job scheduling problem. Our study has applications in energy-aware scheduling, cloud computing, switching cost optimization as well as wavelength assignments in optical networks.