We consider the following fundamental scheduling problem. The input consists of n jobs to be scheduled
on a set of machines of bounded capacities. Each job is associated with a release time, a due
date, a processing time and demand for machine capacity. The goal is to schedule all of the jobs
non-preemptively in their release-time-deadline windows, subject to machine capacity constraints,
such that the total busy time of the machines is minimized. Our problem has important applications
in power-aware scheduling, optical network design and customer service systems. Scheduling to
minimize busy times is APX-hard already in the special case where all jobs have the same (unit)
processing times and can be scheduled in a fixed time interval.
Our main result is a 5-approximation algorithm for general instances. We extend this result to obtain
an algorithm with the same approximation ratio for the problem of scheduling moldable jobs, that
requires also to determine, for each job, one of several processing-time vs. demand configurations.
Better bounds and exact algorithms are derived for several special cases, including proper interval
graphs, intervals forming a clique and laminar families of intervals.