NP-completeness of the Active Time Scheduling Problem

In this paper, we study the active time scheduling problem. We are given n jobs with integral processing times each of which has an integral release time and deadline. The goal is to schedule all the jobs on a machine that can work on b jobs simultaneously, and the objective is to minimize the number of time slots for which the machine is active. The active time scheduling model was introduced by Chang et al. in the context of energy-efficient scheduling. Surprisingly, despite the development of a number of constant factor approximation algorithms for the problem, the complexity of this fundamental problem had remained open. In this paper, we resolve this open problem and show that the active time scheduling problem is indeed NP-complete.